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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 9943 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
| 2 | 0ex 5306 | . . . . . 6 ⊢ ∅ ∈ V | |
| 3 | 2 | snid 4661 | . . . . 5 ⊢ ∅ ∈ {∅} | 
| 4 | opelxpi 5721 | . . . . 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 6920 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
| 8 | 1st2nd2 8054 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
| 9 | xp1st 8047 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
| 10 | elsni 4642 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
| 11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) | 
| 12 | 11 | opeq1d 4878 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈∅, (2nd ‘𝑦)〉) | 
| 13 | 8, 12 | eqtrd 2776 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈∅, (2nd ‘𝑦)〉) | 
| 14 | 13 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (〈∅, 𝑥〉 = 𝑦 ↔ 〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉)) | 
| 15 | eqcom 2743 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈∅, 𝑥〉) | |
| 16 | eqid 2736 | . . . . . . 7 ⊢ ∅ = ∅ | |
| 17 | vex 3483 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 18 | 2, 17 | opth 5480 | . . . . . . 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 7688 | . 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V)) | 
| 24 | 23 | mptru 1546 | 1 ⊢ inl:V–1-1-onto→({∅} × V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 {csn 4625 〈cop 4631 × cxp 5682 –1-1-onto→wf1o 6559 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 inlcinl 9940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1st 8015 df-2nd 8016 df-inl 9943 | 
| This theorem is referenced by: inlresf 9955 inlresf1 9956 djuin 9959 djuun 9967 | 
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