Theorem en3d 7055

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
en3d.1	|- ( ph -> A e. V )
en3d.2	|- ( ph -> B e. W )
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)   V( x, y)   W( x, y)

Proof of Theorem en3d

Step	Hyp	Ref	Expression
1		en3d.1	. 2 |- ( ph -> A e. V )
2		en3d.2	. 2 |- ( ph -> B e. W )
3		eqid 2238	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
4		en3d.3	. . . 4 |- ( ph -> ( x e. A -> C e. B ) )
5	4	imp 124	. . 3 |- ( ( ph /\ x e. A ) -> C e. B )
6		en3d.4	. . . 4 |- ( ph -> ( y e. B -> D e. A ) )
7	6	imp 124	. . 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 124	. . 3 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
10	3, 5, 7, 9	f1o2d 6295	. 2 |- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g 7041	. 2 |- ( ( A e. V /\ B e. W /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1, 2, 10, 11	syl3anc 1278	1 |- ( ph -> A ~~ B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   |-> cmpt 4192   -1-1-onto->wf1o 5376   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-en 7023

This theorem is used by:  en3i  7057  fundmen  7094  mapen  7146  mapxpen  7148  mapunen  7151  ssenen  7152  fzen  10447  uzennn  10873  hashfibclem  11282  hashfacen  11284  hashf1lem1  11285  hashdvds  12999