Theorem en2d 6767

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
en2d.1	|- ( ph -> A e. _V )
en2d.2	|- ( ph -> B e. _V )
en2d.3	|- ( ph -> ( x e. A -> C e. _V ) )
en2d.4	|- ( ph -> ( y e. B -> D e. _V ) )
en2d.5	|- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )

Assertion

Ref	Expression
en2d	|- ( 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 en2d

Step	Hyp	Ref	Expression
1		en2d.1	. 2 |- ( ph -> A e. _V )
2		en2d.2	. 2 |- ( ph -> B e. _V )
3		eqid 2177	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en2d.3	. . . 4 |- ( ph -> ( x e. A -> C e. _V ) )
5	4	imp 124	. . 3 |- ( ( ph /\ x e. A ) -> C e. _V )
6		en2d.4	. . . 4 |- ( ph -> ( y e. B -> D e. _V ) )
7	6	imp 124	. . 3 |- ( ( ph /\ y e. B ) -> D e. _V )
8		en2d.5	. . 3 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
9	3, 5, 7, 8	f1od 6073	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
10		f1oen2g 6754	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
11	1, 2, 9, 10	syl3anc 1238	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   class class class wbr 4003   |-> cmpt 4064   -1-1-onto->wf1o 5215   ~~ cen 6737

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-en 6740

This theorem is referenced by:  en2i  6769  map1  6811