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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 9183 | . . 3 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1onn 8120 | . . . . . 6 ⊢ 1o ∈ ω | |
3 | snidg 4508 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 5485 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) | |
6 | 4, 5 | mpan 686 | . . . 4 ⊢ (𝑥 ∈ V → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
7 | 6 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
8 | fvexd 6558 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V) | |
9 | 1st2nd2 7589 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 7582 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o}) | |
11 | elsni 4493 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o) |
13 | 12 | opeq1d 4720 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈1o, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2831 | . . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈1o, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2805 | . . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (〈1o, 𝑥〉 = 𝑦 ↔ 〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉)) |
16 | eqcom 2802 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈1o, 𝑥〉) | |
17 | eqid 2795 | . . . . . . 7 ⊢ 1o = 1o | |
18 | 1oex 7966 | . . . . . . . 8 ⊢ 1o ∈ V | |
19 | vex 3440 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 18, 19 | opth 5265 | . . . . . . 7 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦))) |
21 | 17, 20 | mpbiran 705 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
22 | 15, 16, 21 | 3bitr3g 314 | . . . . 5 ⊢ (𝑦 ∈ ({1o} × V) → (𝑦 = 〈1o, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
23 | 22 | bicomd 224 | . . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
24 | 23 | ad2antll 725 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
25 | 1, 7, 8, 24 | f1o2d 7262 | . 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V)) |
26 | 25 | mptru 1529 | 1 ⊢ inr:V–1-1-onto→({1o} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ⊤wtru 1523 ∈ wcel 2081 Vcvv 3437 {csn 4476 〈cop 4482 × cxp 5446 –1-1-onto→wf1o 6229 ‘cfv 6230 ωcom 7441 1st c1st 7548 2nd c2nd 7549 1oc1o 7951 inrcinr 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3710 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-om 7442 df-1st 7550 df-2nd 7551 df-1o 7958 df-inr 9183 |
This theorem is referenced by: inrresf 9196 inrresf1 9197 djuin 9198 djuun 9206 |
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