Theorem joindef 17004

Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.)

Hypotheses

Ref	Expression
joindef.u	|- U = ( lub ` K )
joindef.j	|- .\/ = ( join ` K )
joindef.k	|- ( ph -> K e. V )
joindef.x	|- ( ph -> X e. W )
joindef.y	|- ( ph -> Y e. Z )

Assertion

Ref	Expression
joindef	|- ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom U ) )

Proof of Theorem joindef

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		joindef.k	. . 3 |- ( ph -> K e. V )
2		joindef.u	. . . . 5 |- U = ( lub ` K )
3		joindef.j	. . . . 5 |- .\/ = ( join ` K )
4	2, 3	joindm 17003	. . . 4 |- ( K e. V -> dom .\/ = { <. x , y >. | { x , y } e. dom U } )
5	4	eleq2d 2687	. . 3 |- ( K e. V -> ( <. X , Y >. e. dom .\/ <-> <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } ) )
6	1, 5	syl 17	. 2 |- ( ph -> ( <. X , Y >. e. dom .\/ <-> <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } ) )
7		joindef.x	. . 3 |- ( ph -> X e. W )
8		joindef.y	. . 3 |- ( ph -> Y e. Z )
9		preq1 4268	. . . . 5 |- ( x = X -> { x , y } = { X , y } )
10	9	eleq1d 2686	. . . 4 |- ( x = X -> ( { x , y } e. dom U <-> { X , y } e. dom U ) )
11		preq2 4269	. . . . 5 |- ( y = Y -> { X , y } = { X , Y } )
12	11	eleq1d 2686	. . . 4 |- ( y = Y -> ( { X , y } e. dom U <-> { X , Y } e. dom U ) )
13	10, 12	opelopabg 4993	. . 3 |- ( ( X e. W /\ Y e. Z ) -> ( <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } <-> { X , Y } e. dom U ) )
14	7, 8, 13	syl2anc 693	. 2 |- ( ph -> ( <. X , Y >. e. { <. x , y >. | { x , y } e. dom U } <-> { X , Y } e. dom U ) )
15	6, 14	bitrd 268	1 |- ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom U ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cpr 4179   <.cop 4183   {copab 4712   dom cdm 5114   ` cfv 5888   lubclub 16942   joincjn 16944

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-oprab 6654  df-lub 16974  df-join 16976

This theorem is referenced by:  joinval  17005  joincl  17006  joindmss  17007  joineu  17010  clatl  17116