Theorem conrel2d 43682

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 5695	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
2	1	ineq2i 4216	. . . 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 5915	. . . 4 ⊢ (𝜑 → dom ◡𝐴 = dom ∅)
6	5	ineq2d 4219	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ◡𝐴) = (dom 𝐵 ∩ dom ∅))
7		dm0 5930	. . . . . 6 ⊢ dom ∅ = ∅
8	7	ineq2i 4216	. . . . 5 ⊢ (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9		in0 4394	. . . . 5 ⊢ (dom 𝐵 ∩ ∅) = ∅
10	8, 9	eqtri 2764	. . . 4 ⊢ (dom 𝐵 ∩ dom ∅) = ∅
11	10	a1i 11	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
12	3, 6, 11	3eqtrd 2780	. 2 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
13	12	coemptyd 15019	1 ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ∩ cin 3949  ∅c0 4332  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   ∘ ccom 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696

This theorem is referenced by: (None)