Theorem djuf1olemr 7345

Description: Lemma for djulf1or 7347 and djurf1or 7348. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7344. (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 5034	. . 3 |- ( F |` A ) = ( ( x e. _V |-> <. X , x >. ) |` A )
4		ssv 3260	. . . 4 |- A C_ _V
5		resmpt 5086	. . . 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 2253	. 2 |- ( F |` A ) = ( x e. A |-> <. X , x >. )
8	1, 7	djuf1olem 7344	1 |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   {csn 3689   <.cop 3692   |-> cmpt 4171   X. cxp 4747   |` cres 4751   -1-1-onto->wf1o 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335

This theorem is referenced by:  djulf1or  7347  djurf1or  7348