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Theorem djuf1olem 6987
 Description: Lemma for djulf1o 6992 and djurf1o 6993. (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 3591 . . . . 5 𝑋 ∈ {𝑋}
4 opelxpi 4615 . . . . 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 6108 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (2nd𝑦) ∈ 𝐴)
87adantl 275 . . 3 ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd𝑦) ∈ 𝐴)
9 1st2nd2 6117 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 6107 . . . . . . . . . 10 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) ∈ {𝑋})
11 elsni 3578 . . . . . . . . . 10 ((1st𝑦) ∈ {𝑋} → (1st𝑦) = 𝑋)
1210, 11syl 14 . . . . . . . . 9 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) = 𝑋)
1312opeq1d 3747 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑋, (2nd𝑦)⟩)
149, 13eqtrd 2190 . . . . . . 7 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd𝑦)⟩)
1514eqeq2d 2169 . . . . . 6 (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩))
16 eqcom 2159 . . . . . 6 (⟨𝑋, 𝑥⟩ = 𝑦𝑦 = ⟨𝑋, 𝑥⟩)
17 eqid 2157 . . . . . . 7 𝑋 = 𝑋
18 vex 2715 . . . . . . . 8 𝑥 ∈ V
192, 18opth 4196 . . . . . . 7 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ (𝑋 = 𝑋𝑥 = (2nd𝑦)))
2017, 19mpbiran 925 . . . . . 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 6019 . 2 (⊤ → 𝐹:𝐴1-1-onto→({𝑋} × 𝐴))
2524mptru 1344 1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1335  ⊤wtru 1336   ∈ wcel 2128  Vcvv 2712  {csn 3560  ⟨cop 3563   ↦ cmpt 4025   × cxp 4581  –1-1-onto→wf1o 5166  ‘cfv 5167  1st c1st 6080  2nd c2nd 6081 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-1st 6082  df-2nd 6083 This theorem is referenced by:  djuf1olemr  6988  djulf1o  6992  djurf1o  6993
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