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Theorem refreleq 38025
Description: Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
refreleq (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))

Proof of Theorem refreleq
StepHypRef Expression
1 dmeq 5910 . . . . . 6 (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
2 rneq 5942 . . . . . 6 (𝑅 = 𝑆 → ran 𝑅 = ran 𝑆)
31, 2xpeq12d 5713 . . . . 5 (𝑅 = 𝑆 → (dom 𝑅 × ran 𝑅) = (dom 𝑆 × ran 𝑆))
43ineq2d 4214 . . . 4 (𝑅 = 𝑆 → ( I ∩ (dom 𝑅 × ran 𝑅)) = ( I ∩ (dom 𝑆 × ran 𝑆)))
5 id 22 . . . 4 (𝑅 = 𝑆𝑅 = 𝑆)
64, 5sseq12d 4015 . . 3 (𝑅 = 𝑆 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ∩ (dom 𝑆 × ran 𝑆)) ⊆ 𝑆))
7 releq 5782 . . 3 (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
86, 7anbi12d 630 . 2 (𝑅 = 𝑆 → ((( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅) ↔ (( I ∩ (dom 𝑆 × ran 𝑆)) ⊆ 𝑆 ∧ Rel 𝑆)))
9 dfrefrel2 38019 . 2 ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
10 dfrefrel2 38019 . 2 ( RefRel 𝑆 ↔ (( I ∩ (dom 𝑆 × ran 𝑆)) ⊆ 𝑆 ∧ Rel 𝑆))
118, 9, 103bitr4g 313 1 (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  cin 3948  wss 3949   I cid 5579   × cxp 5680  dom cdm 5682  ran crn 5683  Rel wrel 5687   RefRel wrefrel 37687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-refrel 38016
This theorem is referenced by:  eqvreleq  38106
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