Theorem ovmpt2 5636

Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (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 )
ovmpt2.4	|- S e. _V

Assertion

Ref	Expression
ovmpt2	|- ( ( A e. C /\ B e. D ) -> ( 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)

Proof of Theorem ovmpt2

Step	Hyp	Ref	Expression
1		ovmpt2.4	. 2 |- S e. _V
2		ovmpt2g.1	. . 3 |- ( x = A -> R = G )
3		ovmpt2g.2	. . 3 |- ( y = B -> G = S )
4		ovmpt2g.3	. . 3 |- F = ( x e. C , y e. D |-> R )
5	2, 3, 4	ovmpt2g 5635	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
6	1, 5	mp3an3 1221	1 |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557  (class class class)co 5512   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  ixxval  8765  fzval  8876