Theorem djuf1olem 7251

Description: Lemma for djulf1o 7256 and djurf1o 7257. (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 3700	. . . . 5 |- X e. { X }
4		opelxpi 4757	. . . . 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 6328	. . . 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 6337	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
10		xp1st 6327	. . . . . . . . . 10 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) e. { X } )
11		elsni 3687	. . . . . . . . . 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 3868	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. X , ( 2nd ` y ) >. )
14	9, 13	eqtrd 2264	. . . . . . 7 |- ( y e. ( { X } X. A ) -> y = <. X , ( 2nd ` y ) >. )
15	14	eqeq2d 2243	. . . . . 6 |- ( y e. ( { X } X. A ) -> ( <. X , x >. = y <-> <. X , x >. = <. X , ( 2nd ` y ) >. ) )
16		eqcom 2233	. . . . . 6 |- ( <. X , x >. = y <-> y = <. X , x >. )
17		eqid 2231	. . . . . . 7 |- X = X
18		vex 2805	. . . . . . . 8 |- x e. _V
19	2, 18	opth 4329	. . . . . . 7 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> ( X = X /\ x = ( 2nd ` y ) ) )
20	17, 19	mpbiran 948	. . . . . 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 6227	. 2 |- ( T. -> F : A -1-1-onto-> ( { X } X. A ) )
25	24	mptru 1406	1 |- F : A -1-1-onto-> ( { X } X. A )

Syntax hints:   <-> wb 105   = wceq 1397   T. wtru 1398   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   |-> cmpt 4150   X. cxp 4723   -1-1-onto->wf1o 5325   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  djuf1olemr  7252  djulf1o  7256  djurf1o  7257