Theorem elovmpod 6252

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6253 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 6181	. 2 |- ( ph -> ( X O Y ) = D )
9	8	eleq2d 2302	1 |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203  (class class class)co 6050   e. cmpo 6052

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055

This theorem is referenced by: (None)