Theorem coeq0i 30279

Description: coeq0 5509 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 5730	. . . . . 6 |- ( B : E --> F -> ran B C_ F )
2	1	3ad2ant2 1013	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ran B C_ F )
3		sslin 3719	. . . . 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 5728	. . . . . . 7 |- ( A : C --> D -> dom A = C )
6	5	3ad2ant1 1012	. . . . . 6 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> dom A = C )
7	6	ineq1d 3694	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( dom A i^i F ) = ( C i^i F ) )
8		simp3 993	. . . . 5 |- ( ( A : C --> D /\ B : E --> F /\ ( C i^i F ) = (/) ) -> ( C i^i F ) = (/) )
9	7, 8	eqtrd 2503	. . . 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 3811	. . 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 5509	. 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 968   = wceq 1374   i^i cin 3470   C_ wss 3471   (/)c0 3780   dom cdm 4994   ran crn 4995   o. ccom 4998   -->wf 5577

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 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-fn 5584  df-f 5585

This theorem is referenced by:  diophren  30340