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Theorem cores2 6274

Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

**Assertion**
Ref	Expression
cores2	⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

**Proof of Theorem cores2**
Step	Hyp	Ref	Expression
1		dfdm4 5884	. . . . . 6 ⊢ dom 𝐴 = ran ^◡𝐴
2	1	sseq1i 3941	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ^◡𝐴 ⊆ 𝐶)
3		cores 6263	. . . . 5 ⊢ (ran ^◡𝐴 ⊆ 𝐶 → ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = (^◡𝐵 ∘ ^◡𝐴))
4	2, 3	sylbi 220	. . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = (^◡𝐵 ∘ ^◡𝐴))
5		cnvco 5874	. . . . 5 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
6		cocnvcnv1 6272	. . . . 5 ⊢ (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
7	5, 6	eqtri 2786	. . . 4 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
8		cnvco 5874	. . . 4 ⊢ ^◡(𝐴 ∘ 𝐵) = (^◡𝐵 ∘ ^◡𝐴)
9	4, 7, 8	3eqtr4g 2823	. . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡(𝐴 ∘ 𝐵))
10	9	cnveqd 5860	. 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡^◡(𝐴 ∘ 𝐵))
11		relco 6116	. . 3 ⊢ Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
12		dfrel2 6193	. . 3 ⊢ (Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) ↔ ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)))
13	11, 12	mpbi 233	. 2 ⊢ ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
14		relco 6116	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
15		dfrel2 6193	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵))
16	14, 15	mpbi 233	. 2 ⊢ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
17	10, 13, 16	3eqtr3g 2821	1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ⊆ wss 3900 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 Rel wrel 5660

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5252 ax-pr 5395

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 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-br 5107 df-opab 5171 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: fcoi1 6778 ofco2 22796 cycpmconjvlem 33772

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