Theorem en3d 6796

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 2189	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en3d.3	. . . 4 |- ( ph -> ( x e. A -> C e. B ) )
5	4	imp 124	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
6		en3d.4	. . . 4 |- ( ph -> ( y e. B -> D e. A ) )
7	6	imp 124	. . 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 124	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
10	3, 5, 7, 9	f1o2d 6100	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g 6782	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1, 2, 10, 11	syl3anc 1249	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   _Vcvv 2752   class class class wbr 4018   |-> cmpt 4079   -1-1-onto->wf1o 5234   ~~ cen 6765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-en 6768

This theorem is referenced by:  en3i  6798  fundmen  6833  mapen  6875  mapxpen  6877  ssenen  6880  fzen  10075  uzennn  10469  hashfacen  10851  hashdvds  12256