Theorem f1o2d 6310

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 6309	. 2 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
6	5	simpld 459	1 |- ( ph -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   e. cmpt 4348   `'ccnv 4837   -1-1-onto->wf1o 5415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423

This theorem is referenced by:  f1opw2  6311  en3d  7344  f1opwfi  7613  mapfien  7655  mapfienOLD  7925  fin23lem22  8494  incexclem  13297  grplmulf1o  15598  conjghm  15775  gapm  15822  psrbagconf1o  17442  hmeoimaf1o  19341  itg1mulc  21180  resinf1o  21990  eff1olem  22002  sqff1o  22518  dvdsflip  22520  dvdsppwf1o  22524  dvdsflf1o  22525  fcobij  26023  hashgcdlem  29562