Theorem djuf1olem 7051

Description: Lemma for djulf1o 7056 and djurf1o 7057. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)

Hypotheses

Ref	Expression
djuf1olem.1	|- X e. _V
djuf1olem.2	|- F = ( x e. A |-> <. X , x >. )

Assertion

Ref	Expression
djuf1olem	|- F : A -1-1-onto-> ( { X } X. A )

Distinct variable groups:   x, X   x, A

Allowed substitution hint:   F( x)

Proof of Theorem djuf1olem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djuf1olem.2	. . 3 |- F = ( x e. A |-> <. X , x >. )
2		djuf1olem.1	. . . . . 6 |- X e. _V
3	2	snid 3623	. . . . 5 |- X e. { X }
4		opelxpi 4658	. . . . 5 |- ( ( X e. { X } /\ x e. A ) -> <. X , x >. e. ( { X } X. A ) )
5	3, 4	mpan 424	. . . 4 |- ( x e. A -> <. X , x >. e. ( { X } X. A ) )
6	5	adantl 277	. . 3 |- ( ( T. /\ x e. A ) -> <. X , x >. e. ( { X } X. A ) )
7		xp2nd 6166	. . . 4 |- ( y e. ( { X } X. A ) -> ( 2nd ` y ) e. A )
8	7	adantl 277	. . 3 |- ( ( T. /\ y e. ( { X } X. A ) ) -> ( 2nd ` y ) e. A )
9		1st2nd2 6175	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
10		xp1st 6165	. . . . . . . . . 10 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) e. { X } )
11		elsni 3610	. . . . . . . . . 10 |- ( ( 1st ` y ) e. { X } -> ( 1st ` y ) = X )
12	10, 11	syl 14	. . . . . . . . 9 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) = X )
13	12	opeq1d 3784	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. X , ( 2nd ` y ) >. )
14	9, 13	eqtrd 2210	. . . . . . 7 |- ( y e. ( { X } X. A ) -> y = <. X , ( 2nd ` y ) >. )
15	14	eqeq2d 2189	. . . . . 6 |- ( y e. ( { X } X. A ) -> ( <. X , x >. = y <-> <. X , x >. = <. X , ( 2nd ` y ) >. ) )
16		eqcom 2179	. . . . . 6 |- ( <. X , x >. = y <-> y = <. X , x >. )
17		eqid 2177	. . . . . . 7 |- X = X
18		vex 2740	. . . . . . . 8 |- x e. _V
19	2, 18	opth 4237	. . . . . . 7 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> ( X = X /\ x = ( 2nd ` y ) ) )
20	17, 19	mpbiran 940	. . . . . 6 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> x = ( 2nd ` y ) )
21	15, 16, 20	3bitr3g 222	. . . . 5 |- ( y e. ( { X } X. A ) -> ( y = <. X , x >. <-> x = ( 2nd ` y ) ) )
22	21	bicomd 141	. . . 4 |- ( y e. ( { X } X. A ) -> ( x = ( 2nd ` y ) <-> y = <. X , x >. ) )
23	22	ad2antll 491	. . 3 |- ( ( T. /\ ( x e. A /\ y e. ( { X } X. A ) ) ) -> ( x = ( 2nd ` y ) <-> y = <. X , x >. ) )
24	1, 6, 8, 23	f1o2d 6075	. 2 |- ( T. -> F : A -1-1-onto-> ( { X } X. A ) )
25	24	mptru 1362	1 |- F : A -1-1-onto-> ( { X } X. A )

Syntax hints:   <-> wb 105   = wceq 1353   T. wtru 1354   e. wcel 2148   _Vcvv 2737   {csn 3592   <.cop 3595   |-> cmpt 4064   X. cxp 4624   -1-1-onto->wf1o 5215   ` cfv 5216   1stc1st 6138   2ndc2nd 6139

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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141

This theorem is referenced by:  djuf1olemr  7052  djulf1o  7056  djurf1o  7057