Metamath Proof Explorer


Theorem keridl

Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011)

Ref Expression
Hypotheses keridl.1 𝐺 = ( 1st𝑆 )
keridl.2 𝑍 = ( GId ‘ 𝐺 )
Assertion keridl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 keridl.1 𝐺 = ( 1st𝑆 )
2 keridl.2 𝑍 = ( GId ‘ 𝐺 )
3 cnvimass ( 𝐹 “ { 𝑍 } ) ⊆ dom 𝐹
4 eqid ( 1st𝑅 ) = ( 1st𝑅 )
5 eqid ran ( 1st𝑅 ) = ran ( 1st𝑅 )
6 eqid ran 𝐺 = ran 𝐺
7 4 5 1 6 rngohomf ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 : ran ( 1st𝑅 ) ⟶ ran 𝐺 )
8 3 7 fssdm ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st𝑅 ) )
9 eqid ( GId ‘ ( 1st𝑅 ) ) = ( GId ‘ ( 1st𝑅 ) )
10 4 5 9 rngo0cl ( 𝑅 ∈ RingOps → ( GId ‘ ( 1st𝑅 ) ) ∈ ran ( 1st𝑅 ) )
11 10 3ad2ant1 ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( GId ‘ ( 1st𝑅 ) ) ∈ ran ( 1st𝑅 ) )
12 4 9 1 2 rngohom0 ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) = 𝑍 )
13 fvex ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) ∈ V
14 13 elsn ( ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) = 𝑍 )
15 12 14 sylibr ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) ∈ { 𝑍 } )
16 ffn ( 𝐹 : ran ( 1st𝑅 ) ⟶ ran 𝐺𝐹 Fn ran ( 1st𝑅 ) )
17 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( ( GId ‘ ( 1st𝑅 ) ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( GId ‘ ( 1st𝑅 ) ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) ∈ { 𝑍 } ) ) )
18 7 16 17 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( GId ‘ ( 1st𝑅 ) ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( GId ‘ ( 1st𝑅 ) ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( GId ‘ ( 1st𝑅 ) ) ) ∈ { 𝑍 } ) ) )
19 11 15 18 mpbir2and ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( GId ‘ ( 1st𝑅 ) ) ∈ ( 𝐹 “ { 𝑍 } ) )
20 an4 ( ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) ↔ ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ∧ ( ( 𝐹𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) )
21 4 5 1 rngohomadd ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) 𝐺 ( 𝐹𝑦 ) ) )
22 21 adantr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) ∧ ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) 𝐺 ( 𝐹𝑦 ) ) )
23 oveq12 ( ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) → ( ( 𝐹𝑥 ) 𝐺 ( 𝐹𝑦 ) ) = ( 𝑍 𝐺 𝑍 ) )
24 23 adantl ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) ∧ ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) ) → ( ( 𝐹𝑥 ) 𝐺 ( 𝐹𝑦 ) ) = ( 𝑍 𝐺 𝑍 ) )
25 1 rngogrpo ( 𝑆 ∈ RingOps → 𝐺 ∈ GrpOp )
26 6 2 grpoidcl ( 𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺 )
27 6 2 grpolid ( ( 𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
28 25 26 27 syl2anc2 ( 𝑆 ∈ RingOps → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
29 28 3ad2ant2 ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
30 29 ad2antrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) ∧ ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 )
31 22 24 30 3eqtrd ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) ∧ ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) = 𝑍 )
32 31 ex ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) → ( ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) → ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) = 𝑍 ) )
33 fvex ( 𝐹𝑥 ) ∈ V
34 33 elsn ( ( 𝐹𝑥 ) ∈ { 𝑍 } ↔ ( 𝐹𝑥 ) = 𝑍 )
35 fvex ( 𝐹𝑦 ) ∈ V
36 35 elsn ( ( 𝐹𝑦 ) ∈ { 𝑍 } ↔ ( 𝐹𝑦 ) = 𝑍 )
37 34 36 anbi12i ( ( ( 𝐹𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ↔ ( ( 𝐹𝑥 ) = 𝑍 ∧ ( 𝐹𝑦 ) = 𝑍 ) )
38 fvex ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ V
39 38 elsn ( ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) = 𝑍 )
40 32 37 39 3imtr4g ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ) → ( ( ( 𝐹𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) → ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) )
41 40 imdistanda ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ∧ ( ( 𝐹𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
42 4 5 rngogcl ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) )
43 42 3expib ( 𝑅 ∈ RingOps → ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ) )
44 43 3ad2ant1 ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ) )
45 44 anim1d ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) → ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
46 41 45 syld ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑦 ∈ ran ( 1st𝑅 ) ) ∧ ( ( 𝐹𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
47 20 46 syl5bi ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
48 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ) )
49 7 16 48 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ) )
50 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( 𝑦 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) )
51 7 16 50 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( 𝑦 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) )
52 49 51 anbi12d ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ∧ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ) ↔ ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑦 ) ∈ { 𝑍 } ) ) ) )
53 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
54 7 16 53 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) )
55 47 52 54 3imtr4d ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ∧ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ) → ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ) )
56 55 impl ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ) ∧ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ) → ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) )
57 56 ralrimiva ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ) → ∀ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) )
58 34 anbi2i ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) )
59 eqid ( 2nd𝑅 ) = ( 2nd𝑅 )
60 4 59 5 rngocl ( ( 𝑅 ∈ RingOps ∧ 𝑧 ∈ ran ( 1st𝑅 ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) )
61 60 3expb ( ( 𝑅 ∈ RingOps ∧ ( 𝑧 ∈ ran ( 1st𝑅 ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) )
62 61 3ad2antl1 ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑧 ∈ ran ( 1st𝑅 ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) )
63 62 anass1rs ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) )
64 63 adantlrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) )
65 eqid ( 2nd𝑆 ) = ( 2nd𝑆 )
66 4 5 59 65 rngohommul ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑧 ∈ ran ( 1st𝑅 ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) )
67 66 anass1rs ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) )
68 67 adantlrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) )
69 oveq2 ( ( 𝐹𝑥 ) = 𝑍 → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) )
70 69 adantl ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) )
71 70 ad2antlr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) ( 𝐹𝑥 ) ) = ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) )
72 4 5 1 6 rngohomcl ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹𝑧 ) ∈ ran 𝐺 )
73 2 6 1 65 rngorz ( ( 𝑆 ∈ RingOps ∧ ( 𝐹𝑧 ) ∈ ran 𝐺 ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) = 𝑍 )
74 73 3ad2antl2 ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝐹𝑧 ) ∈ ran 𝐺 ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) = 𝑍 )
75 72 74 syldan ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) = 𝑍 )
76 75 adantlr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝐹𝑧 ) ( 2nd𝑆 ) 𝑍 ) = 𝑍 )
77 68 71 76 3eqtrd ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) = 𝑍 )
78 fvex ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ V
79 78 elsn ( ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) = 𝑍 )
80 77 79 sylibr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ { 𝑍 } )
81 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) )
82 7 16 81 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) )
83 82 ad2antrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) )
84 64 80 83 mpbir2and ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) )
85 4 59 5 rngocl ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) )
86 85 3expb ( ( 𝑅 ∈ RingOps ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) )
87 86 3ad2antl1 ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) )
88 87 anassrs ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) )
89 88 adantlrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) )
90 4 5 59 65 rngohommul ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) = ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
91 90 anassrs ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) = ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
92 91 adantlrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) = ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
93 oveq1 ( ( 𝐹𝑥 ) = 𝑍 → ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
94 93 adantl ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) → ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
95 94 ad2antlr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝐹𝑥 ) ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) )
96 2 6 1 65 rngolz ( ( 𝑆 ∈ RingOps ∧ ( 𝐹𝑧 ) ∈ ran 𝐺 ) → ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = 𝑍 )
97 96 3ad2antl2 ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝐹𝑧 ) ∈ ran 𝐺 ) → ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = 𝑍 )
98 72 97 syldan ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = 𝑍 )
99 98 adantlr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑍 ( 2nd𝑆 ) ( 𝐹𝑧 ) ) = 𝑍 )
100 92 95 99 3eqtrd ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) = 𝑍 )
101 fvex ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ V
102 101 elsn ( ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) = 𝑍 )
103 100 102 sylibr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ { 𝑍 } )
104 elpreima ( 𝐹 Fn ran ( 1st𝑅 ) → ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) )
105 7 16 104 3syl ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) )
106 105 ad2antrr ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ran ( 1st𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) )
107 89 103 106 mpbir2and ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) )
108 84 107 jca ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st𝑅 ) ) → ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) )
109 108 ralrimiva ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) ) → ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) )
110 109 ex ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) = 𝑍 ) → ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) )
111 58 110 syl5bi ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st𝑅 ) ∧ ( 𝐹𝑥 ) ∈ { 𝑍 } ) → ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) )
112 49 111 sylbid ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) → ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) )
113 112 imp ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ) → ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) )
114 57 113 jca ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ) → ( ∀ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) )
115 114 ralrimiva ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ∀ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) )
116 4 59 5 9 isidl ( 𝑅 ∈ RingOps → ( ( 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) ↔ ( ( 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st𝑅 ) ∧ ( GId ‘ ( 1st𝑅 ) ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) ) ) )
117 116 3ad2ant1 ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) ↔ ( ( 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st𝑅 ) ∧ ( GId ‘ ( 1st𝑅 ) ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑥 ∈ ( 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st𝑅 ) 𝑦 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st𝑅 ) ( ( 𝑧 ( 2nd𝑅 ) 𝑥 ) ∈ ( 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd𝑅 ) 𝑧 ) ∈ ( 𝐹 “ { 𝑍 } ) ) ) ) ) )
118 8 19 115 117 mpbir3and ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) )