Theorem cocnvres 5268

Description: Restricting a relation and a converse relation when they are composed together. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
cocnvres	⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))

Proof of Theorem cocnvres

Step	Hyp	Ref	Expression
1		resss 5043	. . . 4 ⊢ (𝑆 ↾ dom 𝑅) ⊆ 𝑆
2		dmss 4936	. . . 4 ⊢ ((𝑆 ↾ dom 𝑅) ⊆ 𝑆 → dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆)
3	1, 2	ax-mp 5	. . 3 ⊢ dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆
4		cores2 5256	. . 3 ⊢ (dom (𝑆 ↾ dom 𝑅) ⊆ dom 𝑆 → ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅))
5	3, 4	ax-mp 5	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅)
6		rescnvcnv 5206	. . . 4 ⊢ (◡◡𝑅 ↾ dom 𝑆) = (𝑅 ↾ dom 𝑆)
7	6	cnveqi 4911	. . 3 ⊢ ◡(◡◡𝑅 ↾ dom 𝑆) = ◡(𝑅 ↾ dom 𝑆)
8	7	coeq2i 4896	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))
9		dfdm4 4929	. . . 4 ⊢ dom 𝑅 = ran ◡𝑅
10	9	eqimss2i 3285	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
11		cores 5247	. . 3 ⊢ (ran ◡𝑅 ⊆ dom 𝑅 → ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅) = (𝑆 ∘ ◡𝑅))
12	10, 11	ax-mp 5	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡𝑅) = (𝑆 ∘ ◡𝑅)
13	5, 8, 12	3eqtr3ri 2261	1 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))

Syntax hints:   = wceq 1398   ⊆ wss 3201  ^◡ccnv 4730  dom cdm 4731  ran crn 4732   ↾ cres 4733   ∘ ccom 4735

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743

This theorem is referenced by:  cocnvss  5269