Theorem en3d 6735

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 2165	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en3d.3	. . . 4 |- ( ph -> ( x e. A -> C e. B ) )
5	4	imp 123	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
6		en3d.4	. . . 4 |- ( ph -> ( y e. B -> D e. A ) )
7	6	imp 123	. . 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 123	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
10	3, 5, 7, 9	f1o2d 6043	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g 6721	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1, 2, 10, 11	syl3anc 1228	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   |-> cmpt 4043   -1-1-onto->wf1o 5187   ~~ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-en 6707

This theorem is referenced by:  en3i  6737  fundmen  6772  mapen  6812  mapxpen  6814  ssenen  6817  fzen  9978  uzennn  10371  hashfacen  10749  hashdvds  12153