Theorem conrel1d 37955

Description: Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)

Hypothesis

Ref	Expression
conrel1d.a	|- ( ph -> `' A = (/) )

Assertion

Ref	Expression
conrel1d	|- ( ph -> ( A o. B ) = (/) )

Proof of Theorem conrel1d

Step	Hyp	Ref	Expression
1		incom 3805	. . 3 |- ( dom A i^i ran B ) = ( ran B i^i dom A )
2		dfdm4 5316	. . . . 5 |- dom A = ran `' A
3		conrel1d.a	. . . . . . 7 |- ( ph -> `' A = (/) )
4	3	rneqd 5353	. . . . . 6 |- ( ph -> ran `' A = ran (/) )
5		rn0 5377	. . . . . 6 |- ran (/) = (/)
6	4, 5	syl6eq 2672	. . . . 5 |- ( ph -> ran `' A = (/) )
7	2, 6	syl5eq 2668	. . . 4 |- ( ph -> dom A = (/) )
8		ineq2 3808	. . . . 5 |- ( dom A = (/) -> ( ran B i^i dom A ) = ( ran B i^i (/) ) )
9		in0 3968	. . . . 5 |- ( ran B i^i (/) ) = (/)
10	8, 9	syl6eq 2672	. . . 4 |- ( dom A = (/) -> ( ran B i^i dom A ) = (/) )
11	7, 10	syl 17	. . 3 |- ( ph -> ( ran B i^i dom A ) = (/) )
12	1, 11	syl5eq 2668	. 2 |- ( ph -> ( dom A i^i ran B ) = (/) )
13	12	coemptyd 13718	1 |- ( ph -> ( A o. B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   (/)c0 3915   `'ccnv 5113   dom cdm 5114   ran crn 5115   o. ccom 5118

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

This theorem is referenced by: (None)