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Theorem djuf1olem 7018
Description: Lemma for djulf1o 7023 and djurf1o 7024. (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 3607 . . . . 5 𝑋 ∈ {𝑋}
4 opelxpi 4636 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
53, 4mpan 421 . . . 4 (𝑥𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
65adantl 275 . . 3 ((⊤ ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7 xp2nd 6134 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (2nd𝑦) ∈ 𝐴)
87adantl 275 . . 3 ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd𝑦) ∈ 𝐴)
9 1st2nd2 6143 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 6133 . . . . . . . . . 10 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) ∈ {𝑋})
11 elsni 3594 . . . . . . . . . 10 ((1st𝑦) ∈ {𝑋} → (1st𝑦) = 𝑋)
1210, 11syl 14 . . . . . . . . 9 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) = 𝑋)
1312opeq1d 3764 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑋, (2nd𝑦)⟩)
149, 13eqtrd 2198 . . . . . . 7 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd𝑦)⟩)
1514eqeq2d 2177 . . . . . 6 (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩))
16 eqcom 2167 . . . . . 6 (⟨𝑋, 𝑥⟩ = 𝑦𝑦 = ⟨𝑋, 𝑥⟩)
17 eqid 2165 . . . . . . 7 𝑋 = 𝑋
18 vex 2729 . . . . . . . 8 𝑥 ∈ V
192, 18opth 4215 . . . . . . 7 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ (𝑋 = 𝑋𝑥 = (2nd𝑦)))
2017, 19mpbiran 930 . . . . . 6 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2115, 16, 203bitr3g 221 . . . . 5 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2221bicomd 140 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
2322ad2antll 483 . . 3 ((⊤ ∧ (𝑥𝐴𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
241, 6, 8, 23f1o2d 6043 . 2 (⊤ → 𝐹:𝐴1-1-onto→({𝑋} × 𝐴))
2524mptru 1352 1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1343  wtru 1344  wcel 2136  Vcvv 2726  {csn 3576  cop 3579  cmpt 4043   × cxp 4602  1-1-ontowf1o 5187  cfv 5188  1st c1st 6106  2nd c2nd 6107
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109
This theorem is referenced by:  djuf1olemr  7019  djulf1o  7023  djurf1o  7024
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