Theorem fvmpopr2d 6059

Description: Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)

Hypotheses

Ref	Expression
fvmpopr2d.1	|- ( ph -> F = ( a e. A , b e. B |-> C ) )
fvmpopr2d.2	|- ( ph -> P = <. a , b >. )
fvmpopr2d.3	|- ( ( ph /\ a e. A /\ b e. B ) -> C e. V )

Assertion

Ref	Expression
fvmpopr2d	|- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )

Distinct variable groups:   A, a, b   B, a, b

Allowed substitution hints:   ph( a, b)   C( a, b)   P( a, b)   F( a, b)   V( a, b)

Proof of Theorem fvmpopr2d

Dummy variables c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 5925	. . 3 |- ( a ( a e. A , b e. B |-> C ) b ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. )
2		fvmpopr2d.1	. . . . 5 |- ( ph -> F = ( a e. A , b e. B |-> C ) )
3	2	3ad2ant1 1020	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> F = ( a e. A , b e. B |-> C ) )
4		fvmpopr2d.2	. . . . 5 |- ( ph -> P = <. a , b >. )
5	4	3ad2ant1 1020	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> P = <. a , b >. )
6	3, 5	fveq12d 5565	. . 3 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) )
7	1, 6	eqtr4id 2248	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = ( F ` P ) )
8		nfcv 2339	. . . . 5 |- F/_ c C
9		nfcv 2339	. . . . 5 |- F/_ d C
10		nfcv 2339	. . . . . 6 |- F/_ a d
11		nfcsb1v 3117	. . . . . 6 |- F/_ a [_ c / a ]_ C
12	10, 11	nfcsbw 3121	. . . . 5 |- F/_ a [_ d / b ]_ [_ c / a ]_ C
13		nfcsb1v 3117	. . . . 5 |- F/_ b [_ d / b ]_ [_ c / a ]_ C
14		csbeq1a 3093	. . . . . 6 |- ( a = c -> C = [_ c / a ]_ C )
15		csbeq1a 3093	. . . . . 6 |- ( b = d -> [_ c / a ]_ C = [_ d / b ]_ [_ c / a ]_ C )
16	14, 15	sylan9eq 2249	. . . . 5 |- ( ( a = c /\ b = d ) -> C = [_ d / b ]_ [_ c / a ]_ C )
17	8, 9, 12, 13, 16	cbvmpo 6001	. . . 4 |- ( a e. A , b e. B |-> C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C )
18	17	oveqi 5935	. . 3 |- ( a ( a e. A , b e. B |-> C ) b ) = ( a ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) b )
19		eqidd 2197	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) )
20		equcom 1720	. . . . . . . 8 |- ( a = c <-> c = a )
21		equcom 1720	. . . . . . . 8 |- ( b = d <-> d = b )
22	20, 21	anbi12i 460	. . . . . . 7 |- ( ( a = c /\ b = d ) <-> ( c = a /\ d = b ) )
23	22, 16	sylbir 135	. . . . . 6 |- ( ( c = a /\ d = b ) -> C = [_ d / b ]_ [_ c / a ]_ C )
24	23	eqcomd 2202	. . . . 5 |- ( ( c = a /\ d = b ) -> [_ d / b ]_ [_ c / a ]_ C = C )
25	24	adantl 277	. . . 4 |- ( ( ( ph /\ a e. A /\ b e. B ) /\ ( c = a /\ d = b ) ) -> [_ d / b ]_ [_ c / a ]_ C = C )
26		simp2 1000	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> a e. A )
27		simp3 1001	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> b e. B )
28		fvmpopr2d.3	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V )
29	19, 25, 26, 27, 28	ovmpod 6050	. . 3 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C ) b ) = C )
30	18, 29	eqtrid 2241	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = C )
31	7, 30	eqtr3d 2231	1 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   [_csb 3084   <.cop 3625   ` cfv 5258  (class class class)co 5922   e. cmpo 5924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927

This theorem is referenced by:  mpomulcn  14802