Theorem djuf1olem 7243

Description: Lemma for djulf1o 7248 and djurf1o 7249. (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 3698	. . . . 5 ⊢ 𝑋 ∈ {𝑋}
4		opelxpi 4755	. . . . 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 6324	. . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴)
8	7	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴)
9		1st2nd2 6333	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
10		xp1st 6323	. . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋})
11		elsni 3685	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋)
13	12	opeq1d 3866	. . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑋, (2nd ‘𝑦)⟩)
14	9, 13	eqtrd 2262	. . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd ‘𝑦)⟩)
15	14	eqeq2d 2241	. . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩))
16		eqcom 2231	. . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨𝑋, 𝑥⟩)
17		eqid 2229	. . . . . . 7 ⊢ 𝑋 = 𝑋
18		vex 2803	. . . . . . . 8 ⊢ 𝑥 ∈ V
19	2, 18	opth 4327	. . . . . . 7 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦)))
20	17, 19	mpbiran 946	. . . . . 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 6223	. 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴))
25	24	mptru 1404	1 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  Vcvv 2800  {csn 3667  ⟨cop 3670   ↦ cmpt 4148   × cxp 4721  –1-1-onto→wf1o 5323  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1st 6298  df-2nd 6299

This theorem is referenced by:  djuf1olemr  7244  djulf1o  7248  djurf1o  7249