Theorem coeq0i 37316

Description: coeq0 5644 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 6053	. . . . . 6 |- ( B : E --> F -> ran B C_ F )
2	1	3ad2ant2 1083	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ran B C_ F )
3		sslin 3839	. . . . 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 6051	. . . . . . 7 |- ( A : C --> D -> dom A = C )
6	5	3ad2ant1 1082	. . . . . 6 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> dom A = C )
7	6	ineq1d 3813	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = ( C i^i F ) )
8		simp3 1063	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( C i^i F ) = (/) )
9	7, 8	eqtrd 2656	. . . 4 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = (/) )
10	4, 9	sseqtrd 3641	. . 3 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) C_ (/) )
11		ss0 3974	. . 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 5644	. 2 |- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) )
14	12, 13	sylibr 224	1 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( A o. B ) = (/) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   i^i cin 3573   C_ wss 3574   (/)c0 3915   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884

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  ax-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  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-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-fn 5891  df-f 5892

This theorem is referenced by:  diophren  37377