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Mirrors > Home > ILE Home > Th. List > djuf1olem | GIF version |
Description: Lemma for djulf1o 7057 and djurf1o 7058. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) |
Ref | Expression |
---|---|
djuf1olem.1 | ⊢ 𝑋 ∈ V |
djuf1olem.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩) |
Ref | Expression |
---|---|
djuf1olem | ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuf1olem.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩) | |
2 | djuf1olem.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
3 | 2 | snid 3624 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
4 | opelxpi 4659 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴)) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴)) |
6 | 5 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⟨𝑋, 𝑥⟩ ∈ ({𝑋} × 𝐴)) |
7 | xp2nd 6167 | . . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴) | |
8 | 7 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴) |
9 | 1st2nd2 6176 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) | |
10 | xp1st 6166 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋}) | |
11 | elsni 3611 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋) | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋) |
13 | 12 | opeq1d 3785 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝑋, (2nd ‘𝑦)⟩) |
14 | 9, 13 | eqtrd 2210 | . . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = ⟨𝑋, (2nd ‘𝑦)⟩) |
15 | 14 | eqeq2d 2189 | . . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (⟨𝑋, 𝑥⟩ = 𝑦 ↔ ⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩)) |
16 | eqcom 2179 | . . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨𝑋, 𝑥⟩) | |
17 | eqid 2177 | . . . . . . 7 ⊢ 𝑋 = 𝑋 | |
18 | vex 2741 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
19 | 2, 18 | opth 4238 | . . . . . . 7 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦))) |
20 | 17, 19 | mpbiran 940 | . . . . . 6 ⊢ (⟨𝑋, 𝑥⟩ = ⟨𝑋, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦)) |
21 | 15, 16, 20 | 3bitr3g 222 | . . . . 5 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = ⟨𝑋, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦))) |
22 | 21 | bicomd 141 | . . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩)) |
23 | 22 | ad2antll 491 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨𝑋, 𝑥⟩)) |
24 | 1, 6, 8, 23 | f1o2d 6076 | . 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)) |
25 | 24 | mptru 1362 | 1 ⊢ 𝐹:𝐴–1-1-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-onto→wf1o 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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