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Theorem djuf1olem 6904
Description: Lemma for djulf1o 6909 and djurf1o 6910. (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 3524 . . . . 5 𝑋 ∈ {𝑋}
4 opelxpi 4539 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
53, 4mpan 418 . . . 4 (𝑥𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
65adantl 273 . . 3 ((⊤ ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7 xp2nd 6030 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (2nd𝑦) ∈ 𝐴)
87adantl 273 . . 3 ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd𝑦) ∈ 𝐴)
9 1st2nd2 6039 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 6029 . . . . . . . . . 10 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) ∈ {𝑋})
11 elsni 3513 . . . . . . . . . 10 ((1st𝑦) ∈ {𝑋} → (1st𝑦) = 𝑋)
1210, 11syl 14 . . . . . . . . 9 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) = 𝑋)
1312opeq1d 3679 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑋, (2nd𝑦)⟩)
149, 13eqtrd 2148 . . . . . . 7 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd𝑦)⟩)
1514eqeq2d 2127 . . . . . 6 (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩))
16 eqcom 2117 . . . . . 6 (⟨𝑋, 𝑥⟩ = 𝑦𝑦 = ⟨𝑋, 𝑥⟩)
17 eqid 2115 . . . . . . 7 𝑋 = 𝑋
18 vex 2661 . . . . . . . 8 𝑥 ∈ V
192, 18opth 4127 . . . . . . 7 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ (𝑋 = 𝑋𝑥 = (2nd𝑦)))
2017, 19mpbiran 907 . . . . . 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 480 . . 3 ((⊤ ∧ (𝑥𝐴𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
241, 6, 8, 23f1o2d 5941 . 2 (⊤ → 𝐹:𝐴1-1-onto→({𝑋} × 𝐴))
2524mptru 1323 1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  wtru 1315  wcel 1463  Vcvv 2658  {csn 3495  cop 3498  cmpt 3957   × cxp 4505  1-1-ontowf1o 5090  cfv 5091  1st c1st 6002  2nd c2nd 6003
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005
This theorem is referenced by:  djuf1olemr  6905  djulf1o  6909  djurf1o  6910
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