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Mirrors > Home > MPE Home > Th. List > djurf1o | Structured version Visualization version GIF version |
Description: The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
Ref | Expression |
---|---|
djurf1o | ⊢ inr:V–1-1-onto→({1o} × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 9941 | . . 3 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1onn 8677 | . . . . . 6 ⊢ 1o ∈ ω | |
3 | snidg 4665 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 5726 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) | |
6 | 4, 5 | mpan 690 | . . . 4 ⊢ (𝑥 ∈ V → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
8 | fvexd 6922 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V) | |
9 | 1st2nd2 8052 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 8045 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o}) | |
11 | elsni 4648 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o) |
13 | 12 | opeq1d 4884 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈1o, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2775 | . . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈1o, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (〈1o, 𝑥〉 = 𝑦 ↔ 〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉)) |
16 | eqcom 2742 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈1o, 𝑥〉) | |
17 | eqid 2735 | . . . . . . 7 ⊢ 1o = 1o | |
18 | 1oex 8515 | . . . . . . . 8 ⊢ 1o ∈ V | |
19 | vex 3482 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 18, 19 | opth 5487 | . . . . . . 7 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦))) |
21 | 17, 20 | mpbiran 709 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
22 | 15, 16, 21 | 3bitr3g 313 | . . . . 5 ⊢ (𝑦 ∈ ({1o} × V) → (𝑦 = 〈1o, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
23 | 22 | bicomd 223 | . . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
24 | 23 | ad2antll 729 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
25 | 1, 7, 8, 24 | f1o2d 7687 | . 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V)) |
26 | 25 | mptru 1544 | 1 ⊢ inr:V–1-1-onto→({1o} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 × cxp 5687 –1-1-onto→wf1o 6562 ‘cfv 6563 ωcom 7887 1st c1st 8011 2nd c2nd 8012 1oc1o 8498 inrcinr 9938 |
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-3or 1087 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-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 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-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-inr 9941 |
This theorem is referenced by: inrresf 9954 inrresf1 9955 djuin 9956 djuun 9964 |
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