Theorem ovmpt2g 6445

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 2490	. 2 |- ( ( x = A /\ y = B ) -> R = S )
4		ovmpt2g.3	. 2 |- F = ( x e. C , y e. D |-> R )
5	3, 4	ovmpt2ga 6440	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ w3a 982   = wceq 1437   e. wcel 1870  (class class class)co 6305   |-> cmpt2 6307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310

This theorem is referenced by:  ovmpt2  6446  mapvalg  7490  pmvalg  7491  cdaval  8598  genpv  9423  shftfval  13112  symgov  16982  frlmipval  19268  bcthlem1  22185  motplusg  24447  elghomlem1OLD  25934  signspval  29229  paddval  33071  tgrpov  34023  erngmul  34081  erngmul-rN  34089  dvamulr  34287  dvavadd  34290  dvhmulr  34362  djavalN  34411  djhval  34674  mendmulr  35752