Theorem ovmpt2g 6419

Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ovmpt2g.1	|- ( x = A -> R = G )
ovmpt2g.2	|- ( y = B -> G = S )
ovmpt2g.3	|- F = ( x e. C , y e. D |-> R )

Assertion

Ref	Expression
ovmpt2g	|- ( ( 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)   G( x, y)   H( x, y)

Proof of Theorem ovmpt2g

Step	Hyp	Ref	Expression
1		ovmpt2g.1	. . 3 |- ( x = A -> R = G )
2		ovmpt2g.2	. . 3 |- ( y = B -> G = S )
3	1, 2	sylan9eq 2528	. 2 |- ( ( x = A /\ y = B ) -> R = S )
4		ovmpt2g.3	. 2 |- F = ( x e. C , y e. D |-> R )
5	3, 4	ovmpt2ga 6414	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1379   e. wcel 1767  (class class class)co 6282   |-> cmpt2 6284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287

This theorem is referenced by:  ovmpt2  6420  mapvalg  7427  pmvalg  7428  cdaval  8546  genpv  9373  shftfval  12860  symgov  16207  frlmipval  18574  bcthlem1  21495  motplusg  23654  elghomlem1  25036  signspval  28146  mendmulr  30742  paddval  34594  tgrpov  35544  erngmul  35602  erngmul-rN  35610  dvamulr  35808  dvavadd  35811  dvhmulr  35883  djavalN  35932  djhval  36195