Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnvref4 Structured version   Visualization version   GIF version

Theorem cnvref4 37219
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)) ↔ 𝑅𝑆))

Proof of Theorem cnvref4
StepHypRef Expression
1 dfrel6 37216 . . . . . . 7 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
21biimpi 215 . . . . . 6 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32dmeqd 5906 . . . . 5 (Rel 𝑅 → dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = dom 𝑅)
42rneqd 5938 . . . . 5 (Rel 𝑅 → ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = ran 𝑅)
53, 4xpeq12d 5708 . . . 4 (Rel 𝑅 → (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (dom 𝑅 × ran 𝑅))
65ineq2d 4213 . . 3 (Rel 𝑅 → (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) = (𝑆 ∩ (dom 𝑅 × ran 𝑅)))
76sseq2d 4015 . 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 5695 . . . 4 Rel (dom 𝑅 × ran 𝑅)
9 relin2 5814 . . . 4 (Rel (dom 𝑅 × ran 𝑅) → Rel (𝑅 ∩ (dom 𝑅 × ran 𝑅)))
10 relssinxpdmrn 37218 . . . 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))
118, 9, 10mp2b 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)
122sseq1d 4014 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑆𝑅𝑆))
1311, 12bitrid 283 . 2 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom (𝑅 ∩ (dom 𝑅 × ran 𝑅)) × ran (𝑅 ∩ (dom 𝑅 × ran 𝑅)))) ↔ 𝑅𝑆))
147, 13bitr3d 281 1 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  cin 3948  wss 3949   × cxp 5675  dom cdm 5677  ran crn 5678  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689
This theorem is referenced by:  dfcnvrefrel4  37402
  Copyright terms: Public domain W3C validator