Theorem coeq0i 35595

Description: coeq0 5344 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Assertion

Ref	Expression
coeq0i	|- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( A o. B ) = (/) )

Proof of Theorem coeq0i

Step	Hyp	Ref	Expression
1		frn 5735	. . . . . 6 |- ( B : E --> F -> ran B C_ F )
2	1	3ad2ant2 1030	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ran B C_ F )
3		sslin 3658	. . . . 5 |- ( ran B C_ F -> ( dom A i^i ran B ) C_ ( dom A i^i F ) )
4	2, 3	syl 17	. . . 4 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) C_ ( dom A i^i F ) )
5		fdm 5733	. . . . . . 7 |- ( A : C --> D -> dom A = C )
6	5	3ad2ant1 1029	. . . . . 6 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> dom A = C )
7	6	ineq1d 3633	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = ( C i^i F ) )
8		simp3 1010	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( C i^i F ) = (/) )
9	7, 8	eqtrd 2485	. . . 4 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = (/) )
10	4, 9	sseqtrd 3468	. . 3 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) C_ (/) )
11		ss0 3765	. . 3 |- ( ( dom A i^i ran B ) C_ (/) -> ( dom A i^i ran B ) = (/) )
12	10, 11	syl 17	. 2 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) = (/) )
13		coeq0 5344	. 2 |- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) )
14	12, 13	sylibr 216	1 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( A o. B ) = (/) )

Syntax hints:   -> wi 4   /\ w3a 985   = wceq 1444   i^i cin 3403   C_ wss 3404   (/)c0 3731   dom cdm 4834   ran crn 4835   o. ccom 4838   -->wf 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-fn 5585  df-f 5586

This theorem is referenced by:  diophren  35656