Theorem djuf1olemr 7053

Description: Lemma for djulf1or 7055 and djurf1or 7056. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7052. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)

Hypotheses

Ref	Expression
djuf1olemr.1	|- X e. _V
djuf1olemr.2	|- F = ( x e. _V |-> <. X , x >. )

Assertion

Ref	Expression
djuf1olemr	|- ( F |` A ) : A -1-1-onto-> ( { X } X. A )

Distinct variable groups:   x, X   x, A

Allowed substitution hint:   F( x)

Proof of Theorem djuf1olemr

Step	Hyp	Ref	Expression
1		djuf1olemr.1	. 2 |- X e. _V
2		djuf1olemr.2	. . . 4 |- F = ( x e. _V |-> <. X , x >. )
3	2	reseq1i 4904	. . 3 |- ( F |` A ) = ( ( x e. _V |-> <. X , x >. ) |` A )
4		ssv 3178	. . . 4 |- A C_ _V
5		resmpt 4956	. . . 4 |- ( A C_ _V -> ( ( x e. _V |-> <. X , x >. ) |` A ) = ( x e. A |-> <. X , x >. ) )
6	4, 5	ax-mp 5	. . 3 |- ( ( x e. _V |-> <. X , x >. ) |` A ) = ( x e. A |-> <. X , x >. )
7	3, 6	eqtri 2198	. 2 |- ( F |` A ) = ( x e. A |-> <. X , x >. )
8	1, 7	djuf1olem 7052	1 |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2738   C_ wss 3130   {csn 3593   <.cop 3596   |-> cmpt 4065   X. cxp 4625   |` cres 4629   -1-1-onto->wf1o 5216

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142

This theorem is referenced by:  djulf1or  7055  djurf1or  7056