Theorem djuf1olem 7114

Description: Lemma for djulf1o 7119 and djurf1o 7120. (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 3650	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
4		opelxpi 4692	. . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
5	3, 4	mpan 424	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
6	5	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7		xp2nd 6221	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴)
8	7	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴)
9		1st2nd2 6230	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 6220	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋})
11		elsni 3637	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋)
13	12	opeq1d 3811	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑋, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2226	. . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2205	. . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩))
16		eqcom 2195	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨𝑋, 𝑥⟩)
17		eqid 2193	. . . . . . 7 ⊢ 𝑋 = 𝑋
18		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	2, 18	opth 4267	. . . . . . 7 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦)))
20	17, 19	mpbiran 942	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦))
21	15, 16, 20	3bitr3g 222	. . . . 5 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦)))
22	21	bicomd 141	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
23	22	ad2antll 491	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
24	1, 6, 8, 23	f1o2d 6125	. 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴))
25	24	mptru 1373	1 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  Vcvv 2760  {csn 3619  ⟨cop 3622   ↦ cmpt 4091   × cxp 4658  –1-1-onto→wf1o 5254  ‘cfv 5255  1^st c1st 6193  2^nd c2nd 6194

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by:  djuf1olemr  7115  djulf1o  7119  djurf1o  7120