Theorem coeq0i 29016

Description: coeq0 29015 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 5562	. . . . . 6 |- ( B : E --> F -> ran B C_ F )
2	1	3ad2ant2 1005	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ran B C_ F )
3		sslin 3573	. . . . 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 5560	. . . . . . 7 |- ( A : C --> D -> dom A = C )
6	5	3ad2ant1 1004	. . . . . 6 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> dom A = C )
7	6	ineq1d 3548	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = ( C i^i F ) )
8		simp3 985	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( C i^i F ) = (/) )
9	7, 8	eqtrd 2473	. . . 4 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = (/) )
10	4, 9	sseqtrd 3389	. . 3 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i ran B ) C_ (/) )
11		ss0 3665	. . 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 29015	. 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 960   = wceq 1364   i^i cin 3324   C_ wss 3325   (/)c0 3634   dom cdm 4836   ran crn 4837   o. ccom 4840   -->wf 5411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-fn 5418  df-f 5419

This theorem is referenced by:  diophren  29077