Theorem conrel1d 44503

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 4155	. . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2		dfdm4 5879	. . . . 5 ⊢ dom 𝐴 = ran ◡𝐴
3		conrel1d.a	. . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅)
4	3	rneqd 5922	. . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅)
5		rn0 5910	. . . . . 6 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2811	. . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅)
7	2, 6	eqtrid 2807	. . . 4 ⊢ (𝜑 → dom 𝐴 = ∅)
8		ineq2 4160	. . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9		in0 4345	. . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅
10	8, 9	eqtrdi 2811	. . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
11	7, 10	syl 18	. . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
12	1, 11	eqtrid 2807	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
13	12	coemptyd 15052	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 3898  ∅c0 4279  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by: (None)