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Mirrors > Home > MPE Home > Th. List > djulf1o | Structured version Visualization version GIF version |
Description: The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
Ref | Expression |
---|---|
djulf1o | ⊢ inl:V–1-1-onto→({∅} × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 9940 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 0ex 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 4667 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | opelxpi 5726 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) | |
5 | 3, 4 | mpan 690 | . . . 4 ⊢ (𝑥 ∈ V → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
6 | 5 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
7 | fvexd 6922 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
8 | 1st2nd2 8052 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
9 | xp1st 8045 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
10 | elsni 4648 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) |
12 | 11 | opeq1d 4884 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈∅, (2nd ‘𝑦)〉) |
13 | 8, 12 | eqtrd 2775 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈∅, (2nd ‘𝑦)〉) |
14 | 13 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (〈∅, 𝑥〉 = 𝑦 ↔ 〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉)) |
15 | eqcom 2742 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈∅, 𝑥〉) | |
16 | eqid 2735 | . . . . . . 7 ⊢ ∅ = ∅ | |
17 | vex 3482 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
18 | 2, 17 | opth 5487 | . . . . . . 7 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦))) |
19 | 16, 18 | mpbiran 709 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
20 | 14, 15, 19 | 3bitr3g 313 | . . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = 〈∅, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
21 | 20 | bicomd 223 | . . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
22 | 21 | ad2antll 729 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
23 | 1, 6, 7, 22 | f1o2d 7687 | . 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V)) |
24 | 23 | mptru 1544 | 1 ⊢ inl:V–1-1-onto→({∅} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 × cxp 5687 –1-1-onto→wf1o 6562 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 inlcinl 9937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1st 8013 df-2nd 8014 df-inl 9940 |
This theorem is referenced by: inlresf 9952 inlresf1 9953 djuin 9956 djuun 9964 |
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