Theorem en3d 6747

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 2170	. . 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 6054	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g 6733	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1, 2, 10, 11	syl3anc 1233	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   |-> cmpt 4050   -1-1-onto->wf1o 5197   ~~ cen 6716

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-en 6719

This theorem is referenced by:  en3i  6749  fundmen  6784  mapen  6824  mapxpen  6826  ssenen  6829  fzen  9999  uzennn  10392  hashfacen  10771  hashdvds  12175