Theorem djuf1olemr 7031

Description: Lemma for djulf1or 7033 and djurf1or 7034. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7030. (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 4887	. . 3 |- ( F |` A ) = ( ( x e. _V |-> <. X , x >. ) |` A )
4		ssv 3169	. . . 4 |- A C_ _V
5		resmpt 4939	. . . 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 2191	. 2 |- ( F |` A ) = ( x e. A |-> <. X , x >. )
8	1, 7	djuf1olem 7030	1 |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {csn 3583   <.cop 3586   |-> cmpt 4050   X. cxp 4609   |` cres 4613   -1-1-onto->wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by:  djulf1or  7033  djurf1or  7034