Theorem f1oeq3d 6134

Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
f1oeq3d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
f1oeq3d	|- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Proof of Theorem f1oeq3d

Step	Hyp	Ref	Expression
1		f1oeq3d.1	. 2 |- ( ph -> A = B )
2		f1oeq3 6129	. 2 |- ( A = B -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
3	1, 2	syl 17	1 |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  resdif  6157  ackbij2lem2  9062  equivestrcsetc  16792  coe1mul2lem2  19638  usgrf1oedg  26099  wlkiswwlks2lem5  26759  eupthres  27075  eupthp1  27076  poimirlem9  33418  sge0f1o  40599  nnfoctbdj  40673