Theorem fvmpopr2d 6056

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 5922	. . 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 5562	. . 3 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) )
7	1, 6	eqtr4id 2245	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = ( F ` P ) )
8		nfcv 2336	. . . . 5 |- F/_ c C
9		nfcv 2336	. . . . 5 |- F/_ d C
10		nfcv 2336	. . . . . 6 |- F/_ a d
11		nfcsb1v 3114	. . . . . 6 |- F/_ a [_ c / a ]_ C
12	10, 11	nfcsbw 3118	. . . . 5 |- F/_ a [_ d / b ]_ [_ c / a ]_ C
13		nfcsb1v 3114	. . . . 5 |- F/_ b [_ d / b ]_ [_ c / a ]_ C
14		csbeq1a 3090	. . . . . 6 |- ( a = c -> C = [_ c / a ]_ C )
15		csbeq1a 3090	. . . . . 6 |- ( b = d -> [_ c / a ]_ C = [_ d / b ]_ [_ c / a ]_ C )
16	14, 15	sylan9eq 2246	. . . . 5 |- ( ( a = c /\ b = d ) -> C = [_ d / b ]_ [_ c / a ]_ C )
17	8, 9, 12, 13, 16	cbvmpo 5998	. . . 4 |- ( a e. A , b e. B |-> C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C )
18	17	oveqi 5932	. . 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 2194	. . . 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 1717	. . . . . . . 8 |- ( a = c <-> c = a )
21		equcom 1717	. . . . . . . 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 2199	. . . . 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 6047	. . 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 2238	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = C )
31	7, 30	eqtr3d 2228	1 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   [_csb 3081   <.cop 3622   ` cfv 5255  (class class class)co 5919   e. cmpo 5921

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924

This theorem is referenced by:  mpomulcn  14745