Theorem ovmpt2ga 6790

Description: Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovmpt2ga.1	|- ( ( x = A /\ y = B ) -> R = S )
ovmpt2ga.2	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpt2ga	|- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   x, S, y

Allowed substitution hints:   R( x, y)   F( x, y)   H( x, y)

Proof of Theorem ovmpt2ga

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( S e. H -> S e. _V )
2		ovmpt2ga.2	. . . 4 |- F = ( x e. C , y e. D |-> R )
3	2	a1i 11	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> F = ( x e. C , y e. D |-> R ) )
4		ovmpt2ga.1	. . . 4 |- ( ( x = A /\ y = B ) -> R = S )
5	4	adantl 482	. . 3 |- ( ( ( A e. C /\ B e. D /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		simp1 1061	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> A e. C )
7		simp2 1062	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> B e. D )
8		simp3 1063	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> S e. _V )
9	3, 5, 6, 7, 8	ovmpt2d 6788	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
10	1, 9	syl3an3 1361	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt2a  6791  ovmpt2g  6795  elovmpt2  6879  offval  6904  offval3  7162  mptmpt2opabbrd  7248  bropopvvv  7255  reps  13517  hashbcval  15706  setsvalg  15887  ressval  15927  restval  16087  sylow1lem4  18016  sylow3lem2  18043  sylow3lem3  18044  lsmvalx  18054  mvrfval  19420  opsrval  19474  marrepfval  20366  marrepval0  20367  marepvfval  20371  marepvval0  20372  cnmpt12  21470  cnmpt22  21477  qtopval  21498  flimval  21767  fclsval  21812  ucnval  22081  stdbdmetval  22319  resvval  29827  ofcfval3  30164  fmulcl  39813