Theorem ovmpt2d 5788

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
ovmpt2d.1	|- ( ph -> F = ( x e. C , y e. D |-> R ) )
ovmpt2d.2	|- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
ovmpt2d.3	|- ( ph -> A e. C )
ovmpt2d.4	|- ( ph -> B e. D )
ovmpt2d.5	|- ( ph -> S e. X )

Assertion

Ref	Expression
ovmpt2d	|- ( ph -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, S, y   ph, x, y

Allowed substitution hints:   C( x, y)   D( x, y)   R( x, y)   F( x, y)   X( x, y)

Proof of Theorem ovmpt2d

Step	Hyp	Ref	Expression
1		ovmpt2d.1	. 2 |- ( ph -> F = ( x e. C , y e. D |-> R ) )
2		ovmpt2d.2	. 2 |- ( ( ph /\ ( x = A /\ y = B ) ) -> R = S )
3		eqidd 2090	. 2 |- ( ( ph /\ x = A ) -> D = D )
4		ovmpt2d.3	. 2 |- ( ph -> A e. C )
5		ovmpt2d.4	. 2 |- ( ph -> B e. D )
6		ovmpt2d.5	. 2 |- ( ph -> S e. X )
7	1, 2, 3, 4, 5, 6	ovmpt2dx 5787	1 |- ( ph -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439  (class class class)co 5668   |-> cmpt2 5670

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-setind 4368

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673

This theorem is referenced by:  ovmpt2ga  5790  sprmpt2  6023  iseqovex  9933  resqrexlemp1rp  10502  resqrexlemfp1  10505  lcmval  11386