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Theorem f1oeq3 5062
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq3 (A = B → (𝐹:𝐶1-1-ontoA𝐹:𝐶1-1-ontoB))

Proof of Theorem f1oeq3
StepHypRef Expression
1 f1eq3 5032 . . 3 (A = B → (𝐹:𝐶1-1A𝐹:𝐶1-1B))
2 foeq3 5047 . . 3 (A = B → (𝐹:𝐶ontoA𝐹:𝐶ontoB))
31, 2anbi12d 442 . 2 (A = B → ((𝐹:𝐶1-1A 𝐹:𝐶ontoA) ↔ (𝐹:𝐶1-1B 𝐹:𝐶ontoB)))
4 df-f1o 4852 . 2 (𝐹:𝐶1-1-ontoA ↔ (𝐹:𝐶1-1A 𝐹:𝐶ontoA))
5 df-f1o 4852 . 2 (𝐹:𝐶1-1-ontoB ↔ (𝐹:𝐶1-1B 𝐹:𝐶ontoB))
63, 4, 53bitr4g 212 1 (A = B → (𝐹:𝐶1-1-ontoA𝐹:𝐶1-1-ontoB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  1-1wf1 4842  ontowfo 4843  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1oeq23  5063  f1oeq123d  5066  f1ores  5084  resdif  5091  f1osng  5110  f1oresrab  5272  isoeq5  5388  isoini2  5401  bren  6164  xpcomf1o  6235  frechashgf1o  8846
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