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Theorem cores2 6261

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 5876	. . . . . 6 ⊢ dom 𝐴 = ran ^◡𝐴
2	1	sseq1i 3936	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ^◡𝐴 ⊆ 𝐶)
3		cores 6250	. . . . 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 5866	. . . . 5 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
6		cocnvcnv1 6259	. . . . 5 ⊢ (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
7	5, 6	eqtri 2783	. . . 4 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
8		cnvco 5866	. . . 4 ⊢ ^◡(𝐴 ∘ 𝐵) = (^◡𝐵 ∘ ^◡𝐴)
9	4, 7, 8	3eqtr4g 2820	. . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡(𝐴 ∘ 𝐵))
10	9	cnveqd 5852	. 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡^◡(𝐴 ∘ 𝐵))
11		relco 6103	. . 3 ⊢ Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
12		dfrel2 6180	. . 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 6103	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
15		dfrel2 6180	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵))
16	14, 15	mpbi 233	. 2 ⊢ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
17	10, 13, 16	3eqtr3g 2818	1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ⊆ wss 3895 ^◡ccnv 5649 dom cdm 5650 ran crn 5651 ↾ cres 5652 ∘ ccom 5654 Rel wrel 5655

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-br 5103 df-opab 5167 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662

This theorem is used by: fcoi1 6756 ofco2 22771 cycpmconjvlem 33748

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