Theorem eninl 7401

Description: Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)

Assertion

Ref	Expression
eninl	|- ( A e. V -> (inl " A ) ~~ A )

Proof of Theorem eninl

Step	Hyp	Ref	Expression
1		djulf1or 7360	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1oeng 7009	. . . 4 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) ) -> A ~~ ( { (/) } X. A ) )
3	1, 2	mpan2 425	. . 3 |- ( A e. V -> A ~~ ( { (/) } X. A ) )
4		df-ima 4767	. . . 4 |- (inl " A ) = ran (inl |` A )
5		dff1o5 5628	. . . . . 6 |- ( (inl |` A ) : A -1-1-onto-> ( { (/) } X. A ) <-> ( (inl |` A ) : A -1-1-> ( { (/) } X. A ) /\ ran (inl |` A ) = ( { (/) } X. A ) ) )
6	1, 5	mpbi 145	. . . . 5 |- ( (inl |` A ) : A -1-1-> ( { (/) } X. A ) /\ ran (inl |` A ) = ( { (/) } X. A ) )
7	6	simpri 113	. . . 4 |- ran (inl |` A ) = ( { (/) } X. A )
8	4, 7	eqtri 2255	. . 3 |- (inl " A ) = ( { (/) } X. A )
9	3, 8	breqtrrdi 4156	. 2 |- ( A e. V -> A ~~ (inl " A ) )
10	9	ensymd 7036	1 |- ( A e. V -> (inl " A ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   (/)c0 3512   {csn 3694   class class class wbr 4114   X. cxp 4752   ran crn 4755   |` cres 4756   "cima 4757   -1-1->wf1 5354   -1-1-onto->wf1o 5356   ~~ cen 6986  inlcinl 7349

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-in1 619  ax-in2 620  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-coll 4230  ax-sep 4233  ax-nul 4241  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-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  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-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-er 6780  df-en 6989  df-inl 7351

This theorem is referenced by:  endjudisj  7530  djuen  7531