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Theorem djuf1olem 7052
Description: Lemma for djulf1o 7057 and djurf1o 7058. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
Hypotheses
Ref Expression
djuf1olem.1 𝑋 ∈ V
djuf1olem.2 𝐹 = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
Assertion
Ref Expression
djuf1olem 𝐹:𝐴1-1-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.
StepHypRef Expression
1 djuf1olem.2 . . 3 𝐹 = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)
2 djuf1olem.1 . . . . . 6 𝑋 ∈ V
32snid 3624 . . . . 5 𝑋 ∈ {𝑋}
4 opelxpi 4659 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
53, 4mpan 424 . . . 4 (𝑥𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
65adantl 277 . . 3 ((⊤ ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7 xp2nd 6167 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (2nd𝑦) ∈ 𝐴)
87adantl 277 . . 3 ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd𝑦) ∈ 𝐴)
9 1st2nd2 6176 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 6166 . . . . . . . . . 10 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) ∈ {𝑋})
11 elsni 3611 . . . . . . . . . 10 ((1st𝑦) ∈ {𝑋} → (1st𝑦) = 𝑋)
1210, 11syl 14 . . . . . . . . 9 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) = 𝑋)
1312opeq1d 3785 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑋, (2nd𝑦)⟩)
149, 13eqtrd 2210 . . . . . . 7 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd𝑦)⟩)
1514eqeq2d 2189 . . . . . 6 (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩))
16 eqcom 2179 . . . . . 6 (⟨𝑋, 𝑥⟩ = 𝑦𝑦 = ⟨𝑋, 𝑥⟩)
17 eqid 2177 . . . . . . 7 𝑋 = 𝑋
18 vex 2741 . . . . . . . 8 𝑥 ∈ V
192, 18opth 4238 . . . . . . 7 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ (𝑋 = 𝑋𝑥 = (2nd𝑦)))
2017, 19mpbiran 940 . . . . . 6 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2115, 16, 203bitr3g 222 . . . . 5 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2221bicomd 141 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
2322ad2antll 491 . . 3 ((⊤ ∧ (𝑥𝐴𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
241, 6, 8, 23f1o2d 6076 . 2 (⊤ → 𝐹:𝐴1-1-onto→({𝑋} × 𝐴))
2524mptru 1362 1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wtru 1354  wcel 2148  Vcvv 2738  {csn 3593  cop 3596  cmpt 4065   × cxp 4625  1-1-ontowf1o 5216  cfv 5217  1st c1st 6139  2nd c2nd 6140
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142
This theorem is referenced by:  djuf1olemr  7053  djulf1o  7057  djurf1o  7058
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