Theorem eninl 7199

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 7158	. . . 4 |- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
2		f1oeng 6848	. . . 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 4688	. . . 4 |- (inl " A ) = ran (inl |` A )
5		dff1o5 5531	. . . . . 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 2226	. . 3 |- (inl " A ) = ( { (/) } X. A )
9	3, 8	breqtrrdi 4086	. 2 |- ( A e. V -> A ~~ (inl " A ) )
10	9	ensymd 6875	1 |- ( A e. V -> (inl " A ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   (/)c0 3460   {csn 3633   class class class wbr 4044   X. cxp 4673   ran crn 4676   |` cres 4677   "cima 4678   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ~~ cen 6825  inlcinl 7147

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-er 6620  df-en 6828  df-inl 7149

This theorem is referenced by:  endjudisj  7322  djuen  7323