Theorem fvmpopr2d 6140

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 6003	. . 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 1042	. . . 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 1042	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> P = <. a , b >. )
6	3, 5	fveq12d 5633	. . 3 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = ( ( a e. A , b e. B |-> C ) ` <. a , b >. ) )
7	1, 6	eqtr4id 2281	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = ( F ` P ) )
8		nfcv 2372	. . . . 5 |- F/_ c C
9		nfcv 2372	. . . . 5 |- F/_ d C
10		nfcv 2372	. . . . . 6 |- F/_ a d
11		nfcsb1v 3157	. . . . . 6 |- F/_ a [_ c / a ]_ C
12	10, 11	nfcsbw 3161	. . . . 5 |- F/_ a [_ d / b ]_ [_ c / a ]_ C
13		nfcsb1v 3157	. . . . 5 |- F/_ b [_ d / b ]_ [_ c / a ]_ C
14		csbeq1a 3133	. . . . . 6 |- ( a = c -> C = [_ c / a ]_ C )
15		csbeq1a 3133	. . . . . 6 |- ( b = d -> [_ c / a ]_ C = [_ d / b ]_ [_ c / a ]_ C )
16	14, 15	sylan9eq 2282	. . . . 5 |- ( ( a = c /\ b = d ) -> C = [_ d / b ]_ [_ c / a ]_ C )
17	8, 9, 12, 13, 16	cbvmpo 6082	. . . 4 |- ( a e. A , b e. B |-> C ) = ( c e. A , d e. B |-> [_ d / b ]_ [_ c / a ]_ C )
18	17	oveqi 6013	. . 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 2230	. . . 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 1752	. . . . . . . 8 |- ( a = c <-> c = a )
21		equcom 1752	. . . . . . . 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 2235	. . . . 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 1022	. . . 4 |- ( ( ph /\ a e. A /\ b e. B ) -> a e. A )
27		simp3 1023	. . . 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 6131	. . 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 2274	. 2 |- ( ( ph /\ a e. A /\ b e. B ) -> ( a ( a e. A , b e. B |-> C ) b ) = C )
31	7, 30	eqtr3d 2264	1 |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   [_csb 3124   <.cop 3669   ` cfv 5317  (class class class)co 6000   e. cmpo 6002

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  mpomulcn  15234