Theorem ovmpt2ga 6206

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 2966	. 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 454	. . 3 |- ( ( ( A e. C /\ B e. D /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		simp1 958	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> A e. C )
7		simp2 959	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> B e. D )
8		simp3 960	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> S e. _V )
9	3, 5, 6, 7, 8	ovmpt2d 6204	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
10	1, 9	syl3an3 1220	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726   _Vcvv 2958  (class class class)co 6084   e. cmpt2 6086

This theorem is referenced by:  ovmpt2a  6207  ovmpt2g  6211  elovmpt2  6294  offval  6315  offval3  6321  bropopvvv  6429  hashbcval  13375  setsvalg  13497  ressval  13521  restval  13659  sylow1lem4  15240  sylow3lem2  15267  sylow3lem3  15268  lsmvalx  15278  mvrfval  16489  opsrval  16540  cnmpt12  17704  cnmpt22  17711  qtopval  17732  flimval  18000  fclsval  18045  ucnval  18312  stdbdmetval  18549  wlkon  21535  trlon  21545  pthon  21580  spthon  21587  ofcfval3  24490  fmulcl  27701  is2wlkonot  28395  is2spthonot  28396  2wlkonot  28397  2spthonot  28398  2wlksot  28399  2spthsot  28400  2wlkonot3v  28407  2spthonot3v  28408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089