Theorem djuf1olem 7018

Description: Lemma for djulf1o 7023 and djurf1o 7024. (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 3607	. . . . 5 |- X e. { X }
4		opelxpi 4636	. . . . 5 |- ( ( X e. { X } /\ x e. A ) -> <. X , x >. e. ( { X } X. A ) )
5	3, 4	mpan 421	. . . 4 |- ( x e. A -> <. X , x >. e. ( { X } X. A ) )
6	5	adantl 275	. . 3 |- ( ( T. /\ x e. A ) -> <. X , x >. e. ( { X } X. A ) )
7		xp2nd 6134	. . . 4 |- ( y e. ( { X } X. A ) -> ( 2nd ` y ) e. A )
8	7	adantl 275	. . 3 |- ( ( T. /\ y e. ( { X } X. A ) ) -> ( 2nd ` y ) e. A )
9		1st2nd2 6143	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
10		xp1st 6133	. . . . . . . . . 10 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) e. { X } )
11		elsni 3594	. . . . . . . . . 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 3764	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. X , ( 2nd ` y ) >. )
14	9, 13	eqtrd 2198	. . . . . . 7 |- ( y e. ( { X } X. A ) -> y = <. X , ( 2nd ` y ) >. )
15	14	eqeq2d 2177	. . . . . 6 |- ( y e. ( { X } X. A ) -> ( <. X , x >. = y <-> <. X , x >. = <. X , ( 2nd ` y ) >. ) )
16		eqcom 2167	. . . . . 6 |- ( <. X , x >. = y <-> y = <. X , x >. )
17		eqid 2165	. . . . . . 7 |- X = X
18		vex 2729	. . . . . . . 8 |- x e. _V
19	2, 18	opth 4215	. . . . . . 7 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> ( X = X /\ x = ( 2nd ` y ) ) )
20	17, 19	mpbiran 930	. . . . . 6 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> x = ( 2nd ` y ) )
21	15, 16, 20	3bitr3g 221	. . . . 5 |- ( y e. ( { X } X. A ) -> ( y = <. X , x >. <-> x = ( 2nd ` y ) ) )
22	21	bicomd 140	. . . 4 |- ( y e. ( { X } X. A ) -> ( x = ( 2nd ` y ) <-> y = <. X , x >. ) )
23	22	ad2antll 483	. . 3 |- ( ( T. /\ ( x e. A /\ y e. ( { X } X. A ) ) ) -> ( x = ( 2nd ` y ) <-> y = <. X , x >. ) )
24	1, 6, 8, 23	f1o2d 6043	. 2 |- ( T. -> F : A -1-1-onto-> ( { X } X. A ) )
25	24	mptru 1352	1 |- F : A -1-1-onto-> ( { X } X. A )

Syntax hints:   <-> wb 104   = wceq 1343   T. wtru 1344   e. wcel 2136   _Vcvv 2726   {csn 3576   <.cop 3579   |-> cmpt 4043   X. cxp 4602   -1-1-onto->wf1o 5187   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  djuf1olemr  7019  djulf1o  7023  djurf1o  7024