Theorem elovmpod 6121

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6122 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 6050	. 2 |- ( ph -> ( X O Y ) = D )
9	8	eleq2d 2266	1 |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167  (class class class)co 5922   e. cmpo 5924

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927

This theorem is referenced by: (None)