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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 9845 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩) | |
2 | 0ex 5269 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 4627 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | opelxpi 5675 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V)) | |
5 | 3, 4 | mpan 689 | . . . 4 ⊢ (𝑥 ∈ V → ⟨∅, 𝑥⟩ ∈ ({∅} × V)) |
6 | 5 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → ⟨∅, 𝑥⟩ ∈ ({∅} × V)) |
7 | fvexd 6862 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
8 | 1st2nd2 7965 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩) | |
9 | xp1st 7958 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
10 | elsni 4608 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) |
12 | 11 | opeq1d 4841 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨∅, (2nd ‘𝑦)⟩) |
13 | 8, 12 | eqtrd 2777 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = ⟨∅, (2nd ‘𝑦)⟩) |
14 | 13 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (⟨∅, 𝑥⟩ = 𝑦 ↔ ⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩)) |
15 | eqcom 2744 | . . . . . 6 ⊢ (⟨∅, 𝑥⟩ = 𝑦 ↔ 𝑦 = ⟨∅, 𝑥⟩) | |
16 | eqid 2737 | . . . . . . 7 ⊢ ∅ = ∅ | |
17 | vex 3452 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
18 | 2, 17 | opth 5438 | . . . . . . 7 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦))) |
19 | 16, 18 | mpbiran 708 | . . . . . 6 ⊢ (⟨∅, 𝑥⟩ = ⟨∅, (2nd ‘𝑦)⟩ ↔ 𝑥 = (2nd ‘𝑦)) |
20 | 14, 15, 19 | 3bitr3g 313 | . . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = ⟨∅, 𝑥⟩ ↔ 𝑥 = (2nd ‘𝑦))) |
21 | 20 | bicomd 222 | . . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩)) |
22 | 21 | ad2antll 728 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = ⟨∅, 𝑥⟩)) |
23 | 1, 6, 7, 22 | f1o2d 7612 | . 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V)) |
24 | 23 | mptru 1549 | 1 ⊢ inl:V–1-1-onto→({∅} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 {csn 4591 ⟨cop 4597 × cxp 5636 –1-1-onto→wf1o 6500 ‘cfv 6501 1st c1st 7924 2nd c2nd 7925 inlcinl 9842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1st 7926 df-2nd 7927 df-inl 9845 |
This theorem is referenced by: inlresf 9857 inlresf1 9858 djuin 9861 djuun 9869 |
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