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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 9333 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 0ex 5213 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 4603 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | opelxpi 5594 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) | |
5 | 3, 4 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ V → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
6 | 5 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
7 | fvexd 6687 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
8 | 1st2nd2 7730 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
9 | xp1st 7723 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
10 | elsni 4586 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) |
12 | 11 | opeq1d 4811 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈∅, (2nd ‘𝑦)〉) |
13 | 8, 12 | eqtrd 2858 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈∅, (2nd ‘𝑦)〉) |
14 | 13 | eqeq2d 2834 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (〈∅, 𝑥〉 = 𝑦 ↔ 〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉)) |
15 | eqcom 2830 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈∅, 𝑥〉) | |
16 | eqid 2823 | . . . . . . 7 ⊢ ∅ = ∅ | |
17 | vex 3499 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
18 | 2, 17 | opth 5370 | . . . . . . 7 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦))) |
19 | 16, 18 | mpbiran 707 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
20 | 14, 15, 19 | 3bitr3g 315 | . . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = 〈∅, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
21 | 20 | bicomd 225 | . . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
22 | 21 | ad2antll 727 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
23 | 1, 6, 7, 22 | f1o2d 7401 | . 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 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 {csn 4569 〈cop 4575 × cxp 5555 –1-1-onto→wf1o 6356 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 inlcinl 9330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-1st 7691 df-2nd 7692 df-inl 9333 |
This theorem is referenced by: inlresf 9345 inlresf1 9346 djuin 9349 djuun 9357 |
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