Theorem ovmpt2g 6336

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 2515	. 2 |- ( ( x = A /\ y = B ) -> R = S )
4		ovmpt2g.3	. 2 |- F = ( x e. C , y e. D |-> R )
5	3, 4	ovmpt2ga 6331	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ w3a 965   = wceq 1370   e. wcel 1758  (class class class)co 6201   |-> cmpt2 6203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206

This theorem is referenced by:  ovmpt2  6337  mapvalg  7335  pmvalg  7336  cdaval  8451  genpv  9280  shftfval  12678  symgov  16015  frlmipval  18330  bcthlem1  20968  elghomlem1  24001  signspval  27098  mendmulr  29694  paddval  33781  tgrpov  34731  erngmul  34789  erngmul-rN  34797  dvamulr  34995  dvavadd  34998  dvhmulr  35070  djavalN  35119  djhval  35382