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Theorem cores2 5217

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 4892	. . . . . 6 ⊢ dom 𝐴 = ran ^◡𝐴
2	1	sseq1i 3230	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ^◡𝐴 ⊆ 𝐶)
3		cores 5208	. . . . 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 4884	. . . . 5 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
6		cocnvcnv1 5215	. . . . 5 ⊢ (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
7	5, 6	eqtri 2230	. . . 4 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
8		cnvco 4884	. . . 4 ⊢ ^◡(𝐴 ∘ 𝐵) = (^◡𝐵 ∘ ^◡𝐴)
9	4, 7, 8	3eqtr4g 2267	. . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡(𝐴 ∘ 𝐵))
10	9	cnveqd 4875	. 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡^◡(𝐴 ∘ 𝐵))
11		relco 5203	. . 3 ⊢ Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
12		dfrel2 5155	. . 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 5203	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
15		dfrel2 5155	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵))
16	14, 15	mpbi 145	. 2 ⊢ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
17	10, 13, 16	3eqtr3g 2265	1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1375 ⊆ wss 3177 ^◡ccnv 4695 dom cdm 4696 ran crn 4697 ↾ cres 4698 ∘ ccom 4700 Rel wrel 4701

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272

This theorem depends on definitions: df-bi 117 df-3an 985 df-tru 1378 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ral 2493 df-rex 2494 df-v 2781 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-br 4063 df-opab 4125 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708

This theorem is referenced by: cocnvres 5229 fcoi1 5482

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