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Theorem cores2 5274

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 4947	. . . . . 6 ⊢ dom 𝐴 = ran ^◡𝐴
2	1	sseq1i 3263	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ^◡𝐴 ⊆ 𝐶)
3		cores 5265	. . . . 5 ⊢ (ran ^◡𝐴 ⊆ 𝐶 → ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = (^◡𝐵 ∘ ^◡𝐴))
4	2, 3	sylbi 121	. . . 4 ⊢ (dom 𝐴 ⊆ 𝐶 → ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = (^◡𝐵 ∘ ^◡𝐴))
5		cnvco 4939	. . . . 5 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
6		cocnvcnv1 5272	. . . . 5 ⊢ (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
7	5, 6	eqtri 2253	. . . 4 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
8		cnvco 4939	. . . 4 ⊢ ^◡(𝐴 ∘ 𝐵) = (^◡𝐵 ∘ ^◡𝐴)
9	4, 7, 8	3eqtr4g 2290	. . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡(𝐴 ∘ 𝐵))
10	9	cnveqd 4930	. 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡^◡(𝐴 ∘ 𝐵))
11		relco 5260	. . 3 ⊢ Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
12		dfrel2 5212	. . 3 ⊢ (Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) ↔ ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)))
13	11, 12	mpbi 145	. 2 ⊢ ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
14		relco 5260	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
15		dfrel2 5212	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵))
16	14, 15	mpbi 145	. 2 ⊢ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
17	10, 13, 16	3eqtr3g 2288	1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ⊆ wss 3210 ^◡ccnv 4747 dom cdm 4748 ran crn 4749 ↾ cres 4750 ∘ ccom 4752 Rel wrel 4753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2814 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-br 4109 df-opab 4171 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760

This theorem is referenced by: cocnvres 5286 fcoi1 5546

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