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Theorem cores2 5143

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 4821	. . . . . 6 ⊢ dom 𝐴 = ran ^◡𝐴
2	1	sseq1i 3183	. . . . 5 ⊢ (dom 𝐴 ⊆ 𝐶 ↔ ran ^◡𝐴 ⊆ 𝐶)
3		cores 5134	. . . . 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 4814	. . . . 5 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
6		cocnvcnv1 5141	. . . . 5 ⊢ (^◡^◡(^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
7	5, 6	eqtri 2198	. . . 4 ⊢ ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ((^◡𝐵 ↾ 𝐶) ∘ ^◡𝐴)
8		cnvco 4814	. . . 4 ⊢ ^◡(𝐴 ∘ 𝐵) = (^◡𝐵 ∘ ^◡𝐴)
9	4, 7, 8	3eqtr4g 2235	. . 3 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡(𝐴 ∘ 𝐵))
10	9	cnveqd 4805	. 2 ⊢ (dom 𝐴 ⊆ 𝐶 → ^◡^◡(𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = ^◡^◡(𝐴 ∘ 𝐵))
11		relco 5129	. . 3 ⊢ Rel (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶))
12		dfrel2 5081	. . 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 5129	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
15		dfrel2 5081	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) ↔ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵))
16	14, 15	mpbi 145	. 2 ⊢ ^◡^◡(𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
17	10, 13, 16	3eqtr3g 2233	1 ⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ^◡(^◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1353 ⊆ wss 3131 ^◡ccnv 4627 dom cdm 4628 ran crn 4629 ↾ cres 4630 ∘ ccom 4632 Rel wrel 4633

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-br 4006 df-opab 4067 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640

This theorem is referenced by: cocnvres 5155 fcoi1 5398

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