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Theorem trcleq2lem 13940
 Description: Equality implies bijection. (Contributed by RP, 5-May-2020.)
Assertion
Ref Expression
trcleq2lem (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Proof of Theorem trcleq2lem
StepHypRef Expression
1 sseq2 3776 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
32, 2coeq12d 5424 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐵𝐵))
43, 2sseq12d 3783 . 2 (𝐴 = 𝐵 → ((𝐴𝐴) ⊆ 𝐴 ↔ (𝐵𝐵) ⊆ 𝐵))
51, 4anbi12d 616 1 (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ⊆ wss 3723   ∘ ccom 5254 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-in 3730  df-ss 3737  df-br 4788  df-opab 4848  df-co 5259 This theorem is referenced by:  cvbtrcl  13941  trcleq12lem  13942  trclublem  13944  cotrtrclfv  13961  trclun  13963  trclexi  38451  dftrcl3  38536
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