Theorem eninl 7287

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 7246	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1oeng 6925	. . . 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 4736	. . . 4 |- (inl " A ) = ran (inl |` A )
5		dff1o5 5589	. . . . . 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 2250	. . 3 |- (inl " A ) = ( { (/) } X. A )
9	3, 8	breqtrrdi 4128	. 2 |- ( A e. V -> A ~~ (inl " A ) )
10	9	ensymd 6952	1 |- ( A e. V -> (inl " A ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   (/)c0 3492   {csn 3667   class class class wbr 4086   X. cxp 4721   ran crn 4724   |` cres 4725   "cima 4726   -1-1->wf1 5321   -1-1-onto->wf1o 5323   ~~ cen 6902  inlcinl 7235

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299  df-er 6697  df-en 6905  df-inl 7237

This theorem is referenced by:  endjudisj  7415  djuen  7416