Theorem cnvref4 39325

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 39322	. . . . . . 7 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
2	1	biimpi 219	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	dmeqd 5886	. . . . 5 ⊢ (Rel 𝑅 → dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = dom 𝑅)
4	2	rneqd 5919	. . . . 5 ⊢ (Rel 𝑅 → ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
5	3, 4	xpeq12d 5681	. . . 4 ⊢ (Rel 𝑅 → (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (dom 𝑅 × ran 𝑅))
6	5	ineq2d 4117	. . 3 ⊢ (Rel 𝑅 → (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) = (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
7	6	sseq2d 3940	. 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 5668	. . . 4 ⊢ Rel (dom 𝑅 × ran 𝑅)
9		relin2 5790	. . . 4 ⊢ (Rel (dom 𝑅 × ran 𝑅) → Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
10		relssinxpdmrn 39324	. . . 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 3939	. . 3 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
13	11, 12	bitrid 286	. 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 284	1 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ⊆ wss 3895   ∩ cin 4089   × cxp 5648  dom cdm 5650  ran crn 5651  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  dfcnvrefrel4  39587