Metamath Proof Explorer


Theorem mnurndlem1

Description: Lemma for mnurnd . (Contributed by Rohan Ridenour, 12-Aug-2023)

Ref Expression
Hypotheses mnurndlem1.3 ( 𝜑𝐹 : 𝐴𝑈 )
mnurndlem1.4 𝐴 ∈ V
mnurndlem1.6 ( 𝜑 → ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) )
Assertion mnurndlem1 ( 𝜑 → ran 𝐹𝑤 )

Proof

Step Hyp Ref Expression
1 mnurndlem1.3 ( 𝜑𝐹 : 𝐴𝑈 )
2 mnurndlem1.4 𝐴 ∈ V
3 mnurndlem1.6 ( 𝜑 → ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) )
4 1 ffnd ( 𝜑𝐹 Fn 𝐴 )
5 vex 𝑖 ∈ V
6 5 prid1 𝑖 ∈ { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } }
7 simpr ( ( 𝑖𝐴𝑣 = { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } ) → 𝑣 = { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } )
8 6 7 eleqtrrid ( ( 𝑖𝐴𝑣 = { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } ) → 𝑖𝑣 )
9 eqid ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) = ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } )
10 id ( 𝑖𝐴𝑖𝐴 )
11 prex { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } ∈ V
12 11 a1i ( 𝑖𝐴 → { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } ∈ V )
13 id ( 𝑎 = 𝑖𝑎 = 𝑖 )
14 fveq2 ( 𝑎 = 𝑖 → ( 𝐹𝑎 ) = ( 𝐹𝑖 ) )
15 14 preq1d ( 𝑎 = 𝑖 → { ( 𝐹𝑎 ) , 𝐴 } = { ( 𝐹𝑖 ) , 𝐴 } )
16 13 15 preq12d ( 𝑎 = 𝑖 → { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } = { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } )
17 16 adantl ( ( 𝑖𝐴𝑎 = 𝑖 ) → { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } = { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } )
18 9 10 12 17 rr-elrnmpt3d ( 𝑖𝐴 → { 𝑖 , { ( 𝐹𝑖 ) , 𝐴 } } ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) )
19 8 18 rspcime ( 𝑖𝐴 → ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 )
20 19 rgen 𝑖𝐴𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣
21 ralim ( ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) → ( ∀ 𝑖𝐴𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∀ 𝑖𝐴𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) )
22 3 20 21 mpisyl ( 𝜑 → ∀ 𝑖𝐴𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) )
23 prex { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ V
24 23 rgenw 𝑎𝐴 { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ V
25 eleq2 ( 𝑢 = { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( 𝑖𝑢𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) )
26 unieq ( 𝑢 = { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → 𝑢 = { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } )
27 26 sseq1d ( 𝑢 = { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( 𝑢𝑤 { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) )
28 25 27 anbi12d ( 𝑢 = { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( ( 𝑖𝑢 𝑢𝑤 ) ↔ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) )
29 9 28 rexrnmptw ( ∀ 𝑎𝐴 { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ V → ( ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ↔ ∃ 𝑎𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) )
30 24 29 ax-mp ( ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ↔ ∃ 𝑎𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) )
31 simplrl ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } )
32 simpr ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → 𝑖𝐴 )
33 2 prid2 𝐴 ∈ { ( 𝐹𝑎 ) , 𝐴 }
34 elnotel ( 𝐴 ∈ { ( 𝐹𝑎 ) , 𝐴 } → ¬ { ( 𝐹𝑎 ) , 𝐴 } ∈ 𝐴 )
35 33 34 ax-mp ¬ { ( 𝐹𝑎 ) , 𝐴 } ∈ 𝐴
36 35 a1i ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → ¬ { ( 𝐹𝑎 ) , 𝐴 } ∈ 𝐴 )
37 32 36 elnelneq2d ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → ¬ 𝑖 = { ( 𝐹𝑎 ) , 𝐴 } )
38 elpri ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( 𝑖 = 𝑎𝑖 = { ( 𝐹𝑎 ) , 𝐴 } ) )
39 38 orcomd ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( 𝑖 = { ( 𝐹𝑎 ) , 𝐴 } ∨ 𝑖 = 𝑎 ) )
40 39 ord ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } → ( ¬ 𝑖 = { ( 𝐹𝑎 ) , 𝐴 } → 𝑖 = 𝑎 ) )
41 31 37 40 sylc ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → 𝑖 = 𝑎 )
42 41 fveq2d ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → ( 𝐹𝑖 ) = ( 𝐹𝑎 ) )
43 simplrr ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 )
44 vex 𝑎 ∈ V
45 prex { ( 𝐹𝑎 ) , 𝐴 } ∈ V
46 44 45 unipr { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } = ( 𝑎 ∪ { ( 𝐹𝑎 ) , 𝐴 } )
47 46 sseq1i ( { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ↔ ( 𝑎 ∪ { ( 𝐹𝑎 ) , 𝐴 } ) ⊆ 𝑤 )
48 unss ( ( 𝑎𝑤 ∧ { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 ) ↔ ( 𝑎 ∪ { ( 𝐹𝑎 ) , 𝐴 } ) ⊆ 𝑤 )
49 48 bicomi ( ( 𝑎 ∪ { ( 𝐹𝑎 ) , 𝐴 } ) ⊆ 𝑤 ↔ ( 𝑎𝑤 ∧ { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 ) )
50 49 simprbi ( ( 𝑎 ∪ { ( 𝐹𝑎 ) , 𝐴 } ) ⊆ 𝑤 → { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 )
51 47 50 sylbi ( { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 → { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 )
52 fvex ( 𝐹𝑎 ) ∈ V
53 52 2 prss ( ( ( 𝐹𝑎 ) ∈ 𝑤𝐴𝑤 ) ↔ { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 )
54 53 bicomi ( { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 ↔ ( ( 𝐹𝑎 ) ∈ 𝑤𝐴𝑤 ) )
55 54 simplbi ( { ( 𝐹𝑎 ) , 𝐴 } ⊆ 𝑤 → ( 𝐹𝑎 ) ∈ 𝑤 )
56 43 51 55 3syl ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → ( 𝐹𝑎 ) ∈ 𝑤 )
57 42 56 eqeltrd ( ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) ∧ 𝑖𝐴 ) → ( 𝐹𝑖 ) ∈ 𝑤 )
58 57 ex ( ( 𝑎𝐴 ∧ ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) ) → ( 𝑖𝐴 → ( 𝐹𝑖 ) ∈ 𝑤 ) )
59 58 rexlimiva ( ∃ 𝑎𝐴 ( 𝑖 ∈ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∧ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ⊆ 𝑤 ) → ( 𝑖𝐴 → ( 𝐹𝑖 ) ∈ 𝑤 ) )
60 30 59 sylbi ( ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) → ( 𝑖𝐴 → ( 𝐹𝑖 ) ∈ 𝑤 ) )
61 60 com12 ( 𝑖𝐴 → ( ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) → ( 𝐹𝑖 ) ∈ 𝑤 ) )
62 61 ralimia ( ∀ 𝑖𝐴𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) → ∀ 𝑖𝐴 ( 𝐹𝑖 ) ∈ 𝑤 )
63 22 62 syl ( 𝜑 → ∀ 𝑖𝐴 ( 𝐹𝑖 ) ∈ 𝑤 )
64 fnfvrnss ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑖𝐴 ( 𝐹𝑖 ) ∈ 𝑤 ) → ran 𝐹𝑤 )
65 4 63 64 syl2anc ( 𝜑 → ran 𝐹𝑤 )