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Mirrors > Home > ILE Home > Th. List > djuf1olem | GIF version |
Description: Lemma for djulf1o 6943 and djurf1o 6944. (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 3556 | . . . . 5 ⊢ 𝑋 ∈ {𝑋} |
4 | opelxpi 4571 | . . . . 5 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑥 ∈ 𝐴) → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) | |
5 | 3, 4 | mpan 420 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) |
6 | 5 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 〈𝑋, 𝑥〉 ∈ ({𝑋} × 𝐴)) |
7 | xp2nd 6064 | . . . 4 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (2nd ‘𝑦) ∈ 𝐴) | |
8 | 7 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({𝑋} × 𝐴)) → (2nd ‘𝑦) ∈ 𝐴) |
9 | 1st2nd2 6073 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 6063 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) ∈ {𝑋}) | |
11 | elsni 3545 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {𝑋} → (1st ‘𝑦) = 𝑋) | |
12 | 10, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (1st ‘𝑦) = 𝑋) |
13 | 12 | opeq1d 3711 | . . . . . . . 8 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈𝑋, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2172 | . . . . . . 7 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → 𝑦 = 〈𝑋, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2151 | . . . . . 6 ⊢ (𝑦 ∈ ({𝑋} × 𝐴) → (〈𝑋, 𝑥〉 = 𝑦 ↔ 〈𝑋, 𝑥〉 = 〈𝑋, (2nd ‘𝑦)〉)) |
16 | eqcom 2141 | . . . . . 6 ⊢ (〈𝑋, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈𝑋, 𝑥〉) | |
17 | eqid 2139 | . . . . . . 7 ⊢ 𝑋 = 𝑋 | |
18 | vex 2689 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
19 | 2, 18 | opth 4159 | . . . . . . 7 ⊢ (〈𝑋, 𝑥〉 = 〈𝑋, (2nd ‘𝑦)〉 ↔ (𝑋 = 𝑋 ∧ 𝑥 = (2nd ‘𝑦))) |
20 | 17, 19 | mpbiran 924 | . . . . . 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 482 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ({𝑋} × 𝐴))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈𝑋, 𝑥〉)) |
24 | 1, 6, 8, 23 | f1o2d 5975 | . 2 ⊢ (⊤ → 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)) |
25 | 24 | mptru 1340 | 1 ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 Vcvv 2686 {csn 3527 〈cop 3530 ↦ cmpt 3989 × cxp 4537 –1-1-onto→wf1o 5122 ‘cfv 5123 1st c1st 6036 2nd c2nd 6037 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: djuf1olemr 6939 djulf1o 6943 djurf1o 6944 |
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