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Theorem heeq12 37573
Description: Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
heeq12 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐵))

Proof of Theorem heeq12
StepHypRef Expression
1 simpl 473 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝑅 = 𝑆)
2 simpr 477 . . . 4 ((𝑅 = 𝑆𝐴 = 𝐵) → 𝐴 = 𝐵)
31, 2imaeq12d 5428 . . 3 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅𝐴) = (𝑆𝐵))
43, 2sseq12d 3615 . 2 ((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴) ⊆ 𝐴 ↔ (𝑆𝐵) ⊆ 𝐵))
5 df-he 37570 . 2 (𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
6 df-he 37570 . 2 (𝑆 hereditary 𝐵 ↔ (𝑆𝐵) ⊆ 𝐵)
74, 5, 63bitr4g 303 1 ((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wss 3556  cima 5079   hereditary whe 37569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-he 37570
This theorem is referenced by:  heeq1  37574  heeq2  37575  frege77  37737
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