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Theorem trcleq2lem 14442
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 3903 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
32, 2coeq12d 5707 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐵𝐵))
43, 2sseq12d 3910 . 2 (𝐴 = 𝐵 → ((𝐴𝐴) ⊆ 𝐴 ↔ (𝐵𝐵) ⊆ 𝐵))
51, 4anbi12d 634 1 (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wss 3843  ccom 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-in 3850  df-ss 3860  df-br 5031  df-opab 5093  df-co 5534
This theorem is referenced by:  cvbtrcl  14443  trcleq12lem  14444  trclublem  14446  cotrtrclfv  14463  trclun  14465  trclexi  40795  dftrcl3  40896
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