Theorem conrel2d 43696

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
conrel2d	⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)

Proof of Theorem conrel2d

Step	Hyp	Ref	Expression
1		df-rn 5627	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
2	1	ineq2i 4167	. . . 4 ⊢ (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom ◡𝐴)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = (dom 𝐵 ∩ dom ◡𝐴))
4		conrel1d.a	. . . . 5 ⊢ (𝜑 → ◡𝐴 = ∅)
5	4	dmeqd 5845	. . . 4 ⊢ (𝜑 → dom ◡𝐴 = dom ∅)
6	5	ineq2d 4170	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ◡𝐴) = (dom 𝐵 ∩ dom ∅))
7		dm0 5860	. . . . . 6 ⊢ dom ∅ = ∅
8	7	ineq2i 4167	. . . . 5 ⊢ (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9		in0 4345	. . . . 5 ⊢ (dom 𝐵 ∩ ∅) = ∅
10	8, 9	eqtri 2754	. . . 4 ⊢ (dom 𝐵 ∩ dom ∅) = ∅
11	10	a1i 11	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
12	3, 6, 11	3eqtrd 2770	. 2 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
13	12	coemptyd 14883	1 ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3901  ∅c0 4283  ^◡ccnv 5615  dom cdm 5616  ran crn 5617   ∘ ccom 5620

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628

This theorem is referenced by: (None)