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Theorem djuf1olem 6724
Description: Lemma for djulf1o 6729 and djurf1o 6730. (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 3470 . . . . 5 𝑋 ∈ {𝑋}
4 opelxpi 4459 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
53, 4mpan 415 . . . 4 (𝑥𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
65adantl 271 . . 3 ((⊤ ∧ 𝑥𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴))
7 xp2nd 5919 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (2nd𝑦) ∈ 𝐴)
87adantl 271 . . 3 ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd𝑦) ∈ 𝐴)
9 1st2nd2 5927 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
10 xp1st 5918 . . . . . . . . . 10 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) ∈ {𝑋})
11 elsni 3459 . . . . . . . . . 10 ((1st𝑦) ∈ {𝑋} → (1st𝑦) = 𝑋)
1210, 11syl 14 . . . . . . . . 9 (𝑦 ∈ ({𝑋} × 𝐴) → (1st𝑦) = 𝑋)
1312opeq1d 3623 . . . . . . . 8 (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝑋, (2nd𝑦)⟩)
149, 13eqtrd 2120 . . . . . . 7 (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd𝑦)⟩)
1514eqeq2d 2099 . . . . . 6 (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩))
16 eqcom 2090 . . . . . 6 (⟨𝑋, 𝑥⟩ = 𝑦𝑦 = ⟨𝑋, 𝑥⟩)
17 eqid 2088 . . . . . . 7 𝑋 = 𝑋
18 vex 2622 . . . . . . . 8 𝑥 ∈ V
192, 18opth 4055 . . . . . . 7 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ (𝑋 = 𝑋𝑥 = (2nd𝑦)))
2017, 19mpbiran 886 . . . . . 6 (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd𝑦)⟩ ↔ 𝑥 = (2nd𝑦))
2115, 16, 203bitr3g 220 . . . . 5 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd𝑦)))
2221bicomd 139 . . . 4 (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
2322ad2antll 475 . . 3 ((⊤ ∧ (𝑥𝐴𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩))
241, 6, 8, 23f1o2d 5831 . 2 (⊤ → 𝐹:𝐴1-1-onto→({𝑋} × 𝐴))
2524mptru 1298 1 𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1289  wtru 1290  wcel 1438  Vcvv 2619  {csn 3441  cop 3444  cmpt 3891   × cxp 4426  1-1-ontowf1o 5001  cfv 5002  1st c1st 5891  2nd c2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by:  djuf1olemr  6725  djulf1o  6729  djurf1o  6730
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