Theorem cnvref4 38373

Description: Two ways to say that a relation is a subclass. (Contributed by Peter Mazsa, 11-Apr-2023.)

Assertion

Ref	Expression
cnvref4	⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

Proof of Theorem cnvref4

Step	Hyp	Ref	Expression
1		dfrel6 38370	. . . . . . 7 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
2	1	biimpi 216	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	dmeqd 5890	. . . . 5 ⊢ (Rel 𝑅 → dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = dom 𝑅)
4	2	rneqd 5923	. . . . 5 ⊢ (Rel 𝑅 → ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
5	3, 4	xpeq12d 5690	. . . 4 ⊢ (Rel 𝑅 → (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (dom 𝑅 × ran 𝑅))
6	5	ineq2d 4200	. . 3 ⊢ (Rel 𝑅 → (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) = (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
7	6	sseq2d 3996	. 2 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅))))
8		relxp 5677	. . . 4 ⊢ Rel (dom 𝑅 × ran 𝑅)
9		relin2 5797	. . . 4 ⊢ (Rel (dom 𝑅 × ran 𝑅) → Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
10		relssinxpdmrn 38372	. . . 4 ⊢ (Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)) → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆))
11	8, 9, 10	mp2b 10	. . 3 ⊢ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆)
12	2	sseq1d 3995	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
13	11, 12	bitrid 283	. 2 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ 𝑅 ⊆ 𝑆))
14	7, 13	bitr3d 281	1 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∩ cin 3930   ⊆ wss 3931   × cxp 5657  dom cdm 5659  ran crn 5660  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is referenced by:  dfcnvrefrel4  38555