Theorem f1o2d 6547

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

Hypotheses

Ref	Expression
f1od.1	|- F = ( x e. A |-> C )
f1o2d.2	|- ( ( ph /\ x e. A ) -> C e. B )
f1o2d.3	|- ( ( ph /\ y e. B ) -> D e. A )
f1o2d.4	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )

Assertion

Ref	Expression
f1o2d	|- ( ph -> F : A -1-1-onto-> B )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D   ph, x, y

Allowed substitution hints:   C( x)   D( y)   F( x, y)

Proof of Theorem f1o2d

Step	Hyp	Ref	Expression
1		f1od.1	. . 3 |- F = ( x e. A |-> C )
2		f1o2d.2	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
3		f1o2d.3	. . 3 |- ( ( ph /\ y e. B ) -> D e. A )
4		f1o2d.4	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
5	1, 2, 3, 4	f1ocnv2d 6546	. 2 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
6	5	simpld 465	1 |- ( ph -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   |-> cmpt 4474   `'ccnv 4851   -1-1-onto->wf1o 5599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607

This theorem is referenced by:  f1opw2  6548  en3d  7631  f1opwfi  7903  mapfien  7946  fin23lem22  8782  incexclem  13942  grplmulf1o  16776  conjghm  16961  gapm  17008  psrbagconf1o  18646  hmeoimaf1o  20833  itg1mulc  22710  resinf1o  23533  eff1olem  23545  sqff1o  24157  dvdsflip  24159  dvdsppwf1o  24163  dvdsflf1o  24164  fcobij  28358  hashgcdlem  36118