Theorem conrel1d 41160

Description: Deduction about composition with a class with no relational content. (Contributed by RP, 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 4131	. . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2		dfdm4 5793	. . . . 5 ⊢ dom 𝐴 = ran ◡𝐴
3		conrel1d.a	. . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅)
4	3	rneqd 5836	. . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅)
5		rn0 5824	. . . . . 6 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2795	. . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅)
7	2, 6	syl5eq 2791	. . . 4 ⊢ (𝜑 → dom 𝐴 = ∅)
8		ineq2 4137	. . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9		in0 4322	. . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅
10	8, 9	eqtrdi 2795	. . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
12	1, 11	syl5eq 2791	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
13	12	coemptyd 14618	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ∩ cin 3882  ∅c0 4253  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592

This theorem is referenced by: (None)