Theorem en2d 6940

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 2231	. . 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 6225	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
10		f1oen2g 6927	. 2 |- ( ( A e. _V /\ B e. _V /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
11	1, 2, 9, 10	syl3anc 1273	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   |-> cmpt 4150   -1-1-onto->wf1o 5325   ~~ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6909

This theorem is referenced by:  en2i  6942  map1  6986