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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 9184 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 0ex 5109 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 4512 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | opelxpi 5487 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) | |
5 | 3, 4 | mpan 686 | . . . 4 ⊢ (𝑥 ∈ V → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
6 | 5 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
7 | fvexd 6560 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
8 | 1st2nd2 7591 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
9 | xp1st 7584 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
10 | elsni 4495 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) |
12 | 11 | opeq1d 4722 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈∅, (2nd ‘𝑦)〉) |
13 | 8, 12 | eqtrd 2833 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈∅, (2nd ‘𝑦)〉) |
14 | 13 | eqeq2d 2807 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (〈∅, 𝑥〉 = 𝑦 ↔ 〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉)) |
15 | eqcom 2804 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈∅, 𝑥〉) | |
16 | eqid 2797 | . . . . . . 7 ⊢ ∅ = ∅ | |
17 | vex 3443 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
18 | 2, 17 | opth 5267 | . . . . . . 7 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦))) |
19 | 16, 18 | mpbiran 705 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
20 | 14, 15, 19 | 3bitr3g 314 | . . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = 〈∅, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
21 | 20 | bicomd 224 | . . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
22 | 21 | ad2antll 725 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
23 | 1, 6, 7, 22 | f1o2d 7264 | . 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V)) |
24 | 23 | mptru 1532 | 1 ⊢ inl:V–1-1-onto→({∅} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 Vcvv 3440 ∅c0 4217 {csn 4478 〈cop 4484 × cxp 5448 –1-1-onto→wf1o 6231 ‘cfv 6232 1st c1st 7550 2nd c2nd 7551 inlcinl 9181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-1st 7552 df-2nd 7553 df-inl 9184 |
This theorem is referenced by: inlresf 9196 inlresf1 9197 djuin 9200 djuun 9208 |
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