Theorem cocnvres 5155

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 4933	. . . 4 ⊢ (𝑆 ↾ dom 𝑅) ⊆ 𝑆
2		dmss 4828	. . . 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 5143	. . 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 5093	. . . 4 ⊢ (◡◡𝑅 ↾ dom 𝑆) = (𝑅 ↾ dom 𝑆)
7	6	cnveqi 4804	. . 3 ⊢ ◡(◡◡𝑅 ↾ dom 𝑆) = ◡(𝑅 ↾ dom 𝑆)
8	7	coeq2i 4789	. 2 ⊢ ((𝑆 ↾ dom 𝑅) ∘ ◡(◡◡𝑅 ↾ dom 𝑆)) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))
9		dfdm4 4821	. . . 4 ⊢ dom 𝑅 = ran ◡𝑅
10	9	eqimss2i 3214	. . 3 ⊢ ran ◡𝑅 ⊆ dom 𝑅
11		cores 5134	. . 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 2207	1 ⊢ (𝑆 ∘ ◡𝑅) = ((𝑆 ↾ dom 𝑅) ∘ ◡(𝑅 ↾ dom 𝑆))

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

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:  cocnvss  5156