Theorem en2d 7020

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)

Hypotheses

Ref	Expression
en2d.1	|- ( ph -> A e. V )
en2d.2	|- ( ph -> B e. W )
en2d.3	|- ( ph -> ( x e. A -> C e. X ) )
en2d.4	|- ( ph -> ( y e. B -> D e. Y ) )
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)   V( x, y)   W( x, y)   X( x, y)   Y( x, y)

Proof of Theorem en2d

Step	Hyp	Ref	Expression
1		en2d.1	. 2 |- ( ph -> A e. V )
2		en2d.2	. 2 |- ( ph -> B e. W )
3		eqid 2234	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en2d.3	. . . 4 |- ( ph -> ( x e. A -> C e. X ) )
5	4	imp 124	. . 3 |- ( ( ph /\ x e. A ) -> C e. X )
6		en2d.4	. . . 4 |- ( ph -> ( y e. B -> D e. Y ) )
7	6	imp 124	. . 3 |- ( ( ph /\ y e. B ) -> D e. Y )
8		en2d.5	. . 3 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
9	3, 5, 7, 8	f1od 6266	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
10		f1oen2g 7007	. 2 |- ( ( A e. V /\ B e. W /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
11	1, 2, 9, 10	syl3anc 1274	1 |- ( ph -> A ~~ B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   class class class wbr 4114   |-> cmpt 4176   -1-1-onto->wf1o 5356   ~~ cen 6986

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-en 6989

This theorem is referenced by:  en2i  7022  mapsnend  7065  map1  7067