Theorem en3d 6272

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Hypotheses

Ref	Expression
en3d.1	|- ( ph -> A e. _V )
en3d.2	|- ( ph -> B e. _V )
en3d.3	|- ( ph -> ( x e. A -> C e. B ) )
en3d.4	|- ( ph -> ( y e. B -> D e. A ) )
en3d.5	|- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )

Assertion

Ref	Expression
en3d	|- ( ph -> A ~~ B )

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

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

Proof of Theorem en3d

Step	Hyp	Ref	Expression
1		en3d.1	. 2 |- ( ph -> A e. _V )
2		en3d.2	. 2 |- ( ph -> B e. _V )
3		eqid 2081	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en3d.3	. . . 4 |- ( ph -> ( x e. A -> C e. B ) )
5	4	imp 122	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
6		en3d.4	. . . 4 |- ( ph -> ( y e. B -> D e. A ) )
7	6	imp 122	. . 3 |- ( ( ph /\ y e. B ) -> D e. A )
8		en3d.5	. . . 4 |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )
9	8	imp 122	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
10	3, 5, 7, 9	f1o2d 5725	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g 6258	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1, 2, 10, 11	syl3anc 1169	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   |-> cmpt 3839   -1-1-onto->wf1o 4921   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-en 6245

This theorem is referenced by:  en3i  6274  fundmen  6309  fzen  9062