Theorem fvmpopr2d 6157

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 6020	. . 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 1044	. . . 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 1044	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> P = <. a , b >. )
6	3, 5	fveq12d 5646	. . 3 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) )
7	1, 6	eqtr4id 2283	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = ( F ` P ) )
8		nfcv 2374	. . . . 5 |- F/_ c C
9		nfcv 2374	. . . . 5 |- F/_ d C
10		nfcv 2374	. . . . . 6 |- F/_ a d
11		nfcsb1v 3160	. . . . . 6 |- F/_ a [_ c / a ]_ C
12	10, 11	nfcsbw 3164	. . . . 5 |- F/_ a [_ d / b ]_ [_ c / a ]_ C
13		nfcsb1v 3160	. . . . 5 |- F/_ b [_ d / b ]_ [_ c / a ]_ C
14		csbeq1a 3136	. . . . . 6 |- ( a = c -> C = [_ c / a ]_ C )
15		csbeq1a 3136	. . . . . 6 |- ( b = d -> [_ c / a ]_ C = [_ d / b ]_ [_ c / a ]_ C )
16	14, 15	sylan9eq 2284	. . . . 5 |- ( ( a = c /\ b = d ) -> C = [_ d / b ]_ [_ c / a ]_ C )
17	8, 9, 12, 13, 16	cbvmpo 6099	. . . 4 |- ( a e. A , b e. B |-> C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C )
18	17	oveqi 6030	. . 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 2232	. . . 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 1754	. . . . . . . 8 |- ( a = c <-> c = a )
21		equcom 1754	. . . . . . . 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 2237	. . . . 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 1024	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> a e. A )
27		simp3 1025	. . . 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 6148	. . 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 2276	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = C )
31	7, 30	eqtr3d 2266	1 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   [_csb 3127   <.cop 3672   ` cfv 5326  (class class class)co 6017   e. cmpo 6019

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022

This theorem is referenced by:  mpomulcn  15289