Theorem elovmpt2 6879

Description: Utility lemma for two-parameter classes.

EDITORIAL: can simplify isghm 17660, islmhm 19027. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypotheses

Ref	Expression
elovmpt2.d	|- D = ( a e. A , b e. B |-> C )
elovmpt2.c	|- C e. _V
elovmpt2.e	|- ( ( a = X /\ b = Y ) -> C = E )

Assertion

Ref	Expression
elovmpt2	|- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) )

Distinct variable groups:   A, a, b   B, a, b   E, a, b   F, a, b   X, a, b   Y, a, b

Allowed substitution hints:   C( a, b)   D( a, b)

Proof of Theorem elovmpt2

Step	Hyp	Ref	Expression
1		elovmpt2.d	. . . 4 |- D = ( a e. A , b e. B |-> C )
2	1	elmpt2cl 6876	. . 3 |- ( F e. ( X D Y ) -> ( X e. A /\ Y e. B ) )
3		elovmpt2.c	. . . . . . 7 |- C e. _V
4	3	gen2 1723	. . . . . 6 |- A. a A. b C e. _V
5		elovmpt2.e	. . . . . . . 8 |- ( ( a = X /\ b = Y ) -> C = E )
6	5	eleq1d 2686	. . . . . . 7 |- ( ( a = X /\ b = Y ) -> ( C e. _V <-> E e. _V ) )
7	6	spc2gv 3296	. . . . . 6 |- ( ( X e. A /\ Y e. B ) -> ( A. a A. b C e. _V -> E e. _V ) )
8	4, 7	mpi 20	. . . . 5 |- ( ( X e. A /\ Y e. B ) -> E e. _V )
9	5, 1	ovmpt2ga 6790	. . . . 5 |- ( ( X e. A /\ Y e. B /\ E e. _V ) -> ( X D Y ) = E )
10	8, 9	mpd3an3 1425	. . . 4 |- ( ( X e. A /\ Y e. B ) -> ( X D Y ) = E )
11	10	eleq2d 2687	. . 3 |- ( ( X e. A /\ Y e. B ) -> ( F e. ( X D Y ) <-> F e. E ) )
12	2, 11	biadan2 674	. 2 |- ( F e. ( X D Y ) <-> ( ( X e. A /\ Y e. B ) /\ F e. E ) )
13		df-3an 1039	. 2 |- ( ( X e. A /\ Y e. B /\ F e. E ) <-> ( ( X e. A /\ Y e. B ) /\ F e. E ) )
14	12, 13	bitr4i 267	1 |- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  isgim  17704  oppglsm  18057  islmim  19062