Theorem djuf1olem 7346

Description: Lemma for djulf1o 7351 and djurf1o 7352. (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 3722	. . . . 5 |- X e. { X }
4		opelxpi 4783	. . . . 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 6362	. . . 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 6371	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
10		xp1st 6361	. . . . . . . . . 10 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) e. { X } )
11		elsni 3709	. . . . . . . . . 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 3891	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. X , ( 2nd ` y ) >. )
14	9, 13	eqtrd 2267	. . . . . . 7 |- ( y e. ( { X } X. A ) -> y = <. X , ( 2nd ` y ) >. )
15	14	eqeq2d 2246	. . . . . 6 |- ( y e. ( { X } X. A ) -> ( <. X , x >. = y <-> <. X , x >. = <. X , ( 2nd ` y ) >. ) )
16		eqcom 2236	. . . . . 6 |- ( <. X , x >. = y <-> y = <. X , x >. )
17		eqid 2234	. . . . . . 7 |- X = X
18		vex 2818	. . . . . . . 8 |- x e. _V
19	2, 18	opth 4355	. . . . . . 7 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> ( X = X /\ x = ( 2nd ` y ) ) )
20	17, 19	mpbiran 949	. . . . . 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 6262	. 2 |- ( T. -> F : A -1-1-onto-> ( { X } X. A ) )
25	24	mptru 1407	1 |- F : A -1-1-onto-> ( { X } X. A )

Syntax hints:   <-> wb 105   = wceq 1398   T. wtru 1399   e. wcel 2205   _Vcvv 2815   {csn 3691   <.cop 3694   |-> cmpt 4173   X. cxp 4749   -1-1-onto->wf1o 5353   ` cfv 5354   1stc1st 6334   2ndc2nd 6335

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337

This theorem is referenced by:  djuf1olemr  7347  djulf1o  7351  djurf1o  7352