Theorem ovmpt2ga 6162

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 2924	. 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 453	. . 3 |- ( ( ( A e. C /\ B e. D /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		simp1 957	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> A e. C )
7		simp2 958	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> B e. D )
8		simp3 959	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> S e. _V )
9	3, 5, 6, 7, 8	ovmpt2d 6160	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
10	1, 9	syl3an3 1219	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   _Vcvv 2916  (class class class)co 6040   e. cmpt2 6042

This theorem is referenced by:  ovmpt2a  6163  ovmpt2g  6167  elovmpt2  6250  offval  6271  offval3  6277  bropopvvv  6385  hashbcval  13325  setsvalg  13447  ressval  13471  restval  13609  sylow1lem4  15190  sylow3lem2  15217  sylow3lem3  15218  lsmvalx  15228  mvrfval  16439  opsrval  16490  cnmpt12  17652  cnmpt22  17659  qtopval  17680  flimval  17948  fclsval  17993  ucnval  18260  stdbdmetval  18497  wlkon  21483  trlon  21493  pthon  21528  spthon  21535  ofcfval3  24438  fmulcl  27578  is2wlkonot  28060  is2spthonot  28061  2wlkonot  28062  2spthonot  28063  2wlksot  28064  2spthsot  28065  2wlkonot3v  28072  2spthonot3v  28073

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045