Theorem djuf1olemr 7232

Description: Lemma for djulf1or 7234 and djurf1or 7235. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7231. (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 5001	. . 3 |- ( F |` A ) = ( ( x e. _V |-> <. X , x >. ) |` A )
4		ssv 3246	. . . 4 |- A C_ _V
5		resmpt 5053	. . . 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 2250	. 2 |- ( F |` A ) = ( x e. A |-> <. X , x >. )
8	1, 7	djuf1olem 7231	1 |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666   <.cop 3669   |-> cmpt 4145   X. cxp 4717   |` cres 4721   -1-1-onto->wf1o 5317

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 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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6292  df-2nd 6293

This theorem is referenced by:  djulf1or  7234  djurf1or  7235