Theorem conrel1d 43904

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 4161	. . 3 ⊢ (dom 𝐴 ∩ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
2		dfdm4 5844	. . . . 5 ⊢ dom 𝐴 = ran ◡𝐴
3		conrel1d.a	. . . . . . 7 ⊢ (𝜑 → ◡𝐴 = ∅)
4	3	rneqd 5887	. . . . . 6 ⊢ (𝜑 → ran ◡𝐴 = ran ∅)
5		rn0 5875	. . . . . 6 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → ran ◡𝐴 = ∅)
7	2, 6	eqtrid 2783	. . . 4 ⊢ (𝜑 → dom 𝐴 = ∅)
8		ineq2 4166	. . . . 5 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = (ran 𝐵 ∩ ∅))
9		in0 4347	. . . . 5 ⊢ (ran 𝐵 ∩ ∅) = ∅
10	8, 9	eqtrdi 2787	. . . 4 ⊢ (dom 𝐴 = ∅ → (ran 𝐵 ∩ dom 𝐴) = ∅)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → (ran 𝐵 ∩ dom 𝐴) = ∅)
12	1, 11	eqtrid 2783	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
13	12	coemptyd 14902	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3900  ∅c0 4285  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636

This theorem is referenced by: (None)