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	⊢ (𝜑 → ◡𝐴 = ∅)

Assertion

Ref	Expression
conrel1d	⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Proof of Theorem conrel1d

Step	Hyp	Ref	Expression
1		incom 3805	. . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2		dfdm4 5316	. . . . 5 ⊢ dom 𝐴 = ran ◡𝐴
3		conrel1d.a	. . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅)
4	3	rneqd 5353	. . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅)
5		rn0 5377	. . . . . 6 ⊢ ran ∅ = ∅
6	4, 5	syl6eq 2672	. . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅)
7	2, 6	syl5eq 2668	. . . 4 ⊢ (𝜑 → dom 𝐴 = ∅)
8		ineq2 3808	. . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9		in0 3968	. . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅
10	8, 9	syl6eq 2672	. . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
12	1, 11	syl5eq 2668	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
13	12	coemptyd 13718	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1483   ∩ cin 3573  ∅c0 3915  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ∘ 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)