Metamath Proof Explorer


Theorem satfv0fun

Description: The value of the satisfaction predicate as function over wff codes at (/) is a function. (Contributed by AV, 15-Oct-2023)

Ref Expression
Assertion satfv0fun ( ( 𝑀𝑉𝐸𝑊 ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) )

Proof

Step Hyp Ref Expression
1 funopab ( Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) } ↔ ∀ 𝑥 ∃* 𝑦𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) )
2 oveq1 ( 𝑖 = 𝑘 → ( 𝑖𝑔 𝑗 ) = ( 𝑘𝑔 𝑗 ) )
3 2 eqeq2d ( 𝑖 = 𝑘 → ( 𝑥 = ( 𝑖𝑔 𝑗 ) ↔ 𝑥 = ( 𝑘𝑔 𝑗 ) ) )
4 fveq2 ( 𝑖 = 𝑘 → ( 𝑓𝑖 ) = ( 𝑓𝑘 ) )
5 4 breq1d ( 𝑖 = 𝑘 → ( ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) ↔ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) ) )
6 5 rabbidv ( 𝑖 = 𝑘 → { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } )
7 6 eqeq2d ( 𝑖 = 𝑘 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } ) )
8 3 7 anbi12d ( 𝑖 = 𝑘 → ( ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ( 𝑥 = ( 𝑘𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } ) ) )
9 oveq2 ( 𝑗 = 𝑙 → ( 𝑘𝑔 𝑗 ) = ( 𝑘𝑔 𝑙 ) )
10 9 eqeq2d ( 𝑗 = 𝑙 → ( 𝑥 = ( 𝑘𝑔 𝑗 ) ↔ 𝑥 = ( 𝑘𝑔 𝑙 ) ) )
11 fveq2 ( 𝑗 = 𝑙 → ( 𝑓𝑗 ) = ( 𝑓𝑙 ) )
12 11 breq2d ( 𝑗 = 𝑙 → ( ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) ↔ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) ) )
13 12 rabbidv ( 𝑗 = 𝑙 → { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } )
14 13 eqeq2d ( 𝑗 = 𝑙 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) )
15 10 14 anbi12d ( 𝑗 = 𝑙 → ( ( 𝑥 = ( 𝑘𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) ) )
16 8 15 cbvrex2vw ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) )
17 eqtr2 ( ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑥 = ( 𝑘𝑔 𝑙 ) ) → ( 𝑖𝑔 𝑗 ) = ( 𝑘𝑔 𝑙 ) )
18 goeleq12bg ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑖𝑔 𝑗 ) = ( 𝑘𝑔 𝑙 ) ↔ ( 𝑖 = 𝑘𝑗 = 𝑙 ) ) )
19 4 adantr ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( 𝑓𝑖 ) = ( 𝑓𝑘 ) )
20 19 eqcomd ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( 𝑓𝑘 ) = ( 𝑓𝑖 ) )
21 11 adantl ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( 𝑓𝑗 ) = ( 𝑓𝑙 ) )
22 21 eqcomd ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( 𝑓𝑙 ) = ( 𝑓𝑗 ) )
23 20 22 breq12d ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) ↔ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) ) )
24 23 rabbidv ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } )
25 eqeq12 ( ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → ( 𝑦 = 𝑧 ↔ { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) )
26 24 25 syl5ibrcom ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → 𝑦 = 𝑧 ) )
27 26 expd ( ( 𝑖 = 𝑘𝑗 = 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → 𝑦 = 𝑧 ) ) )
28 18 27 syl6bi ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑖𝑔 𝑗 ) = ( 𝑘𝑔 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → 𝑦 = 𝑧 ) ) ) )
29 17 28 syl5 ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑥 = ( 𝑘𝑔 𝑙 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → 𝑦 = 𝑧 ) ) ) )
30 29 expd ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖𝑔 𝑗 ) → ( 𝑥 = ( 𝑘𝑔 𝑙 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → 𝑦 = 𝑧 ) ) ) ) )
31 30 imp4a ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖𝑔 𝑗 ) → ( ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → 𝑦 = 𝑧 ) ) ) )
32 31 com34 ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( 𝑥 = ( 𝑖𝑔 𝑗 ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } → ( ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → 𝑦 = 𝑧 ) ) ) )
33 32 impd ( ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) ∧ ( 𝑖 ∈ ω ∧ 𝑗 ∈ ω ) ) → ( ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → ( ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → 𝑦 = 𝑧 ) ) )
34 33 rexlimdvva ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → ( ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → 𝑦 = 𝑧 ) ) )
35 34 com23 ( ( 𝑘 ∈ ω ∧ 𝑙 ∈ ω ) → ( ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → 𝑦 = 𝑧 ) ) )
36 35 rexlimivv ( ∃ 𝑘 ∈ ω ∃ 𝑙 ∈ ω ( 𝑥 = ( 𝑘𝑔 𝑙 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑘 ) 𝐸 ( 𝑓𝑙 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → 𝑦 = 𝑧 ) )
37 16 36 sylbi ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) → 𝑦 = 𝑧 ) )
38 37 imp ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ) → 𝑦 = 𝑧 )
39 38 gen2 𝑦𝑧 ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ) → 𝑦 = 𝑧 )
40 eqeq1 ( 𝑦 = 𝑧 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ↔ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) )
41 40 anbi2d ( 𝑦 = 𝑧 → ( ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ) )
42 41 2rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ) )
43 42 mo4 ( ∃* 𝑦𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ↔ ∀ 𝑦𝑧 ( ( ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ∧ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) ) → 𝑦 = 𝑧 ) )
44 39 43 mpbir ∃* 𝑦𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } )
45 1 44 mpgbir Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) }
46 eqid ( 𝑀 Sat 𝐸 ) = ( 𝑀 Sat 𝐸 )
47 46 satfv0 ( ( 𝑀𝑉𝐸𝑊 ) → ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) } )
48 47 funeqd ( ( 𝑀𝑉𝐸𝑊 ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) ↔ Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑖 ∈ ω ∃ 𝑗 ∈ ω ( 𝑥 = ( 𝑖𝑔 𝑗 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ( 𝑓𝑖 ) 𝐸 ( 𝑓𝑗 ) } ) } ) )
49 45 48 mpbiri ( ( 𝑀𝑉𝐸𝑊 ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ ∅ ) )