Theorem elovmpod 6277

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6278 in deduction form. (Revised by AV, 20-Apr-2025.)

Hypotheses

Ref	Expression
elovmpod.o	|- O = ( a e. A , b e. B |-> C )
elovmpod.x	|- ( ph -> X e. A )
elovmpod.y	|- ( ph -> Y e. B )
elovmpod.d	|- ( ph -> D e. V )
elovmpod.c	|- ( ( a = X /\ b = Y ) -> C = D )

Assertion

Ref	Expression
elovmpod	|- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )

Distinct variable groups:   D, a, b   X, a, b   Y, a, b   ph, a, b

Allowed substitution hints:   A( a, b)   B( a, b)   C( a, b)   E( a, b)   O( a, b)   V( a, b)

Proof of Theorem elovmpod

Step	Hyp	Ref	Expression
1		elovmpod.o	. . . 4 |- O = ( a e. A , b e. B |-> C )
2	1	a1i 9	. . 3 |- ( ph -> O = ( a e. A , b e. B |-> C ) )
3		elovmpod.c	. . . 4 |- ( ( a = X /\ b = Y ) -> C = D )
4	3	adantl 277	. . 3 |- ( ( ph /\ ( a = X /\ b = Y ) ) -> C = D )
5		elovmpod.x	. . 3 |- ( ph -> X e. A )
6		elovmpod.y	. . 3 |- ( ph -> Y e. B )
7		elovmpod.d	. . 3 |- ( ph -> D e. V )
8	2, 4, 5, 6, 7	ovmpod 6206	. 2 |- ( ph -> ( X O Y ) = D )
9	8	eleq2d 2308	1 |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209  (class class class)co 6075   e. cmpo 6077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080

This theorem is referenced by: (None)