Theorem eninr 7428

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

Assertion

Ref	Expression
eninr	|- ( A e. V -> (inr " A ) ~~ A )

Proof of Theorem eninr

Step	Hyp	Ref	Expression
1		djurf1or 7387	. . . 4 |- (inr |` A ) : A -1-1-onto-> ( { 1o } X. A )
2		f1oeng 7033	. . . 4 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> ( { 1o } X. A ) ) -> A ~~ ( { 1o } X. A ) )
3	1, 2	mpan2 429	. . 3 |- ( A e. V -> A ~~ ( { 1o } X. A ) )
4		df-ima 4782	. . . 4 |- (inr " A ) = ran (inr |` A )
5		dff1o5 5643	. . . . . 6 |- ( (inr |` A ) : A -1-1-onto-> ( { 1o } X. A ) <-> ( (inr |` A ) : A -1-1-> ( { 1o } X. A ) /\ ran (inr |` A ) = ( { 1o } X. A ) ) )
6	1, 5	mpbi 145	. . . . 5 |- ( (inr |` A ) : A -1-1-> ( { 1o } X. A ) /\ ran (inr |` A ) = ( { 1o } X. A ) )
7	6	simpri 113	. . . 4 |- ran (inr |` A ) = ( { 1o } X. A )
8	4, 7	eqtri 2259	. . 3 |- (inr " A ) = ( { 1o } X. A )
9	3, 8	breqtrrdi 4167	. 2 |- ( A e. V -> A ~~ (inr " A ) )
10	9	ensymd 7060	1 |- ( A e. V -> (inr " A ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3705   class class class wbr 4125   X. cxp 4767   ran crn 4770   |` cres 4771   "cima 4772   -1-1->wf1 5369   -1-1-onto->wf1o 5371   1oc1o 6670   ~~ cen 7010  inrcinr 7376

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 623  ax-in2 624  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-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-er 6797  df-en 7013  df-inr 7378

This theorem is referenced by:  endjudisj  7556  djuen  7557