Theorem coeq0i 30848

Description: coeq0 5522 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 5743	. . . . . 6 |- ( B : E --> F -> ran B C_ F )
2	1	3ad2ant2 1018	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ran B C_ F )
3		sslin 3720	. . . . 5 |- ( ran B C_ F -> ( dom A i^i ran B ) C_ ( dom A i^i F ) )
4	2, 3	syl 16	. . . 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 5741	. . . . . . 7 |- ( A : C --> D -> dom A = C )
6	5	3ad2ant1 1017	. . . . . 6 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> dom A = C )
7	6	ineq1d 3695	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = ( C i^i F ) )
8		simp3 998	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( C i^i F ) = (/) )
9	7, 8	eqtrd 2498	. . . 4 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = (/) )
10	4, 9	sseqtrd 3535	. . 3 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) C_ (/) )
11		ss0 3825	. . 3 |- ( ( dom A i^i ran B ) C_ (/) -> ( dom A i^i ran B ) = (/) )
12	10, 11	syl 16	. 2 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) = (/) )
13		coeq0 5522	. 2 |- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) )
14	12, 13	sylibr 212	1 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( A o. B ) = (/) )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1395   i^i cin 3470   C_ wss 3471   (/)c0 3793   dom cdm 5008   ran crn 5009   o. ccom 5012   -->wf 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-fn 5597  df-f 5598

This theorem is referenced by:  diophren  30909