Theorem djuf1olem 6946

Description: Lemma for djulf1o 6951 and djurf1o 6952. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)

Hypotheses

Ref	Expression
djuf1olem.1	⊢ 𝑋 ∈ V
djuf1olem.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)

Assertion

Ref	Expression
djuf1olem	⊢ 𝐹:𝐴–1-1-onto→({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 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)
2		djuf1olem.1	. . . . . 6 ⊢ 𝑋 ∈ V
3	2	snid 3563	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
4		opelxpi 4579	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
5	3, 4	mpan 421	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
6	5	adantl 275	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7		xp2nd 6072	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴)
8	7	adantl 275	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴)
9		1st2nd2 6081	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 6071	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋})
11		elsni 3550	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋)
13	12	opeq1d 3719	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑋, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2173	. . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2152	. . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩))
16		eqcom 2142	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨𝑋, 𝑥⟩)
17		eqid 2140	. . . . . . 7 ⊢ 𝑋 = 𝑋
18		vex 2692	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	2, 18	opth 4167	. . . . . . 7 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦)))
20	17, 19	mpbiran 925	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
21	15, 16, 20	3bitr3g 221	. . . . 5 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
22	21	bicomd 140	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
23	22	ad2antll 483	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
24	1, 6, 8, 23	f1o2d 5983	. 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴))
25	24	mptru 1341	1 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1332  ⊤wtru 1333   ∈ wcel 1481  Vcvv 2689  {csn 3532  ⟨cop 3535   ↦ cmpt 3997   × cxp 4545  –1-1-onto→wf1o 5130  ‘cfv 5131  1^st c1st 6044  2^nd c2nd 6045

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by:  djuf1olemr  6947  djulf1o  6951  djurf1o  6952