Theorem djuf1olem 7312

Description: Lemma for djulf1o 7317 and djurf1o 7318. (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 3704	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
4		opelxpi 4763	. . . . 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 6338	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴)
8	7	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴)
9		1st2nd2 6347	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 6337	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋})
11		elsni 3691	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋)
13	12	opeq1d 3873	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑋, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2264	. . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2243	. . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩))
16		eqcom 2233	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨𝑋, 𝑥⟩)
17		eqid 2231	. . . . . . 7 ⊢ 𝑋 = 𝑋
18		vex 2806	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	2, 18	opth 4335	. . . . . . 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 ∈ ({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 6238	. 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴))
25	24	mptru 1407	1 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2202  Vcvv 2803  {csn 3673  ⟨cop 3676   ↦ cmpt 4155   × cxp 4729  –1-1-onto→wf1o 5332  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by:  djuf1olemr  7313  djulf1o  7317  djurf1o  7318