Theorem elovmpod 6202

Description: Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6203 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 6131	. 2 |- ( ph -> ( X O Y ) = D )
9	8	eleq2d 2299	1 |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200  (class class class)co 6000   e. cmpo 6002

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005

This theorem is referenced by: (None)