Theorem djuf1olem 7065

Description: Lemma for djulf1o 7070 and djurf1o 7071. (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 3635	. . . . 5 |- X e. { X }
4		opelxpi 4670	. . . . 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 6180	. . . 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 6189	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
10		xp1st 6179	. . . . . . . . . 10 |- ( y e. ( { X } X. A ) -> ( 1st ` y ) e. { X } )
11		elsni 3622	. . . . . . . . . 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 3796	. . . . . . . 8 |- ( y e. ( { X } X. A ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. X , ( 2nd ` y ) >. )
14	9, 13	eqtrd 2220	. . . . . . 7 |- ( y e. ( { X } X. A ) -> y = <. X , ( 2nd ` y ) >. )
15	14	eqeq2d 2199	. . . . . 6 |- ( y e. ( { X } X. A ) -> ( <. X , x >. = y <-> <. X , x >. = <. X , ( 2nd ` y ) >. ) )
16		eqcom 2189	. . . . . 6 |- ( <. X , x >. = y <-> y = <. X , x >. )
17		eqid 2187	. . . . . . 7 |- X = X
18		vex 2752	. . . . . . . 8 |- x e. _V
19	2, 18	opth 4249	. . . . . . 7 |- ( <. X , x >. = <. X , ( 2nd ` y ) >. <-> ( X = X /\ x = ( 2nd ` y ) ) )
20	17, 19	mpbiran 941	. . . . . 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 6089	. 2 |- ( T. -> F : A -1-1-onto-> ( { X } X. A ) )
25	24	mptru 1372	1 |- F : A -1-1-onto-> ( { X } X. A )

Syntax hints:   <-> wb 105   = wceq 1363   T. wtru 1364   e. wcel 2158   _Vcvv 2749   {csn 3604   <.cop 3607   |-> cmpt 4076   X. cxp 4636   -1-1-onto->wf1o 5227   ` cfv 5228   1stc1st 6152   2ndc2nd 6153

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6154  df-2nd 6155

This theorem is referenced by:  djuf1olemr  7066  djulf1o  7070  djurf1o  7071