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Mirrors > Home > ILE Home > Th. List > djuf1olem | GIF version |
Description: Lemma for djulf1o 7023 and djurf1o 7024. (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 3607 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
4 | opelxpi 4636 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑥 ∈ 𝐴) → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) | |
5 | 3, 4 | mpan 421 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) |
6 | 5 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) |
7 | xp2nd 6134 | . . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴) | |
8 | 7 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴) |
9 | 1st2nd2 6143 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 6133 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋}) | |
11 | elsni 3594 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋) | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋) |
13 | 12 | opeq1d 3764 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈𝑋, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2198 | . . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = 〈𝑋, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2177 | . . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (〈𝑋, 𝑥〉 = 𝑦 ↔ 〈𝑋, 𝑥〉 = 〈𝑋, (2nd ‘𝑦)〉)) |
16 | eqcom 2167 | . . . . . 6 ⊢ (〈𝑋, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈𝑋, 𝑥〉) | |
17 | eqid 2165 | . . . . . . 7 ⊢ 𝑋 = 𝑋 | |
18 | vex 2729 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
19 | 2, 18 | opth 4215 | . . . . . . 7 ⊢ (〈𝑋, 𝑥〉 = 〈𝑋, (2nd ‘𝑦)〉 ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦))) |
20 | 17, 19 | mpbiran 930 | . . . . . 6 ⊢ (〈𝑋, 𝑥〉 = 〈𝑋, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
21 | 15, 16, 20 | 3bitr3g 221 | . . . . 5 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑦 = 〈𝑋, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
22 | 21 | bicomd 140 | . . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈𝑋, 𝑥〉)) |
23 | 22 | ad2antll 483 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈𝑋, 𝑥〉)) |
24 | 1, 6, 8, 23 | f1o2d 6043 | . 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)) |
25 | 24 | mptru 1352 | 1 ⊢ 𝐹:𝐴–1-1-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-onto→wf1o 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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