Theorem opeldmd 5327

Description: Membership of first of an ordered pair in a domain. Deduction version of opeldm 5328. (Contributed by AV, 11-Mar-2021.)

Hypotheses

Ref	Expression
opeldmd.1	|- ( ph -> A e. V )
opeldmd.2	|- ( ph -> B e. W )

Assertion

Ref	Expression
opeldmd	|- ( ph -> ( <. A , B >. e. C -> A e. dom C ) )

Proof of Theorem opeldmd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeldmd.2	. . 3 |- ( ph -> B e. W )
2		opeq2 4403	. . . . 5 |- ( y = B -> <. A , y >. = <. A , B >. )
3	2	eleq1d 2686	. . . 4 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	3	spcegv 3294	. . 3 |- ( B e. W -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
5	1, 4	syl 17	. 2 |- ( ph -> ( <. A , B >. e. C -> E. y <. A , y >. e. C ) )
6		opeldmd.1	. . 3 |- ( ph -> A e. V )
7		eldm2g 5320	. . 3 |- ( A e. V -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
8	6, 7	syl 17	. 2 |- ( ph -> ( A e. dom C <-> E. y <. A , y >. e. C ) )
9	5, 8	sylibrd 249	1 |- ( ph -> ( <. A , B >. e. C -> A e. dom C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   dom cdm 5114

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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-br 4654  df-dm 5124

This theorem is referenced by:  eupth2eucrct  27077