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Theorem trcleq2lem 14920
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 4004 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 id 22 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
32, 2coeq12d 5856 . . 3 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐵𝐵))
43, 2sseq12d 4011 . 2 (𝐴 = 𝐵 → ((𝐴𝐴) ⊆ 𝐴 ↔ (𝐵𝐵) ⊆ 𝐵))
51, 4anbi12d 631 1 (𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wss 3944  ccom 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3951  df-ss 3961  df-br 5142  df-opab 5204  df-co 5678
This theorem is referenced by:  cvbtrcl  14921  trcleq12lem  14922  trclublem  14924  cotrtrclfv  14941  trclun  14943  trclexi  42142  dftrcl3  42242
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