Theorem conrel2d 40007

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 5565	. . . . 5 ⊢ ran 𝐴 = dom ◡𝐴
2	1	ineq2i 4185	. . . 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 5773	. . . 4 ⊢ (𝜑 → dom ◡𝐴 = dom ∅)
6	5	ineq2d 4188	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ◡𝐴) = (dom 𝐵 ∩ dom ∅))
7		dm0 5789	. . . . . 6 ⊢ dom ∅ = ∅
8	7	ineq2i 4185	. . . . 5 ⊢ (dom 𝐵 ∩ dom ∅) = (dom 𝐵 ∩ ∅)
9		in0 4344	. . . . 5 ⊢ (dom 𝐵 ∩ ∅) = ∅
10	8, 9	eqtri 2844	. . . 4 ⊢ (dom 𝐵 ∩ dom ∅) = ∅
11	10	a1i 11	. . 3 ⊢ (𝜑 → (dom 𝐵 ∩ dom ∅) = ∅)
12	3, 6, 11	3eqtrd 2860	. 2 ⊢ (𝜑 → (dom 𝐵 ∩ ran 𝐴) = ∅)
13	12	coemptyd 14338	1 ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1533   ∩ cin 3934  ∅c0 4290  ^◡ccnv 5553  dom cdm 5554  ran crn 5555   ∘ ccom 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566

This theorem is referenced by: (None)