Theorem f1oeq2 2796

Description: Equality theorem for one-to-one onto functions.

Assertion

Ref	Expression
f1oeq2	|- (A = B -> (F:A-1-1-onto->C <-> F:B-1-1-onto->C))

Proof of Theorem f1oeq2

Step	Hyp	Ref	Expression
1		f1eq2 2777	. . 3 |- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))
2		foeq2 2785	. . 3 |- (A = B -> (F:A-onto->C <-> F:B-onto->C))
3	1, 2	anbi12d 476	. 2 |- (A = B -> ((F:A-1-1->C /\ F:A-onto->C) <-> (F:B-1-1->C /\ F:B-onto->C)))
4		df-f1o 2437	. 2 |- (F:A-1-1-onto->C <-> (F:A-1-1->C /\ F:A-onto->C))
5		df-f1o 2437	. 2 |- (F:B-1-1-onto->C <-> (F:B-1-1->C /\ F:B-onto->C))
6	3, 4, 5	3bitr4g 428	1 |- (A = B -> (F:A-1-1-onto->C <-> F:B-1-1-onto->C))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421

This theorem is referenced by:  isoeq4 2928  breng 3280  unfilem3 3440  infxpidmlem2 4934  infxpidmlem3 4935  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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