Theorem ovmpt2g 6795

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 2676	. 2 |- ( ( x = A /\ y = B ) -> R = S )
4		ovmpt2g.3	. 2 |- F = ( x e. C , y e. D |-> R )
5	3, 4	ovmpt2ga 6790	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990  (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:  ovmpt2  6796  mapvalg  7867  pmvalg  7868  cdaval  8992  genpv  9821  shftfval  13810  symgov  17810  frlmipval  20118  bcthlem1  23121  motplusg  25437  signspval  30629  elghomlem1OLD  33684  paddval  35084  tgrpov  36036  erngmul  36094  erngmul-rN  36102  dvamulr  36300  dvavadd  36303  dvhmulr  36375  djavalN  36424  djhval  36687  mendmulr  37758