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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 9943 | . . 3 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
| 2 | 1onn 8678 | . . . . . 6 ⊢ 1o ∈ ω | |
| 3 | snidg 4660 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o}) | |
| 4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ {1o} |
| 5 | opelxpi 5722 | . . . . 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 6921 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V) | |
| 9 | 1st2nd2 8053 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
| 10 | xp1st 8046 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o}) | |
| 11 | elsni 4643 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o) | |
| 12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o) |
| 13 | 12 | opeq1d 4879 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈1o, (2nd ‘𝑦)〉) |
| 14 | 9, 13 | eqtrd 2777 | . . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈1o, (2nd ‘𝑦)〉) |
| 15 | 14 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (〈1o, 𝑥〉 = 𝑦 ↔ 〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉)) |
| 16 | eqcom 2744 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈1o, 𝑥〉) | |
| 17 | eqid 2737 | . . . . . . 7 ⊢ 1o = 1o | |
| 18 | 1oex 8516 | . . . . . . . 8 ⊢ 1o ∈ V | |
| 19 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 20 | 18, 19 | opth 5481 | . . . . . . 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 1547 | 1 ⊢ inr:V–1-1-onto→({1o} × V) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 × cxp 5683 –1-1-onto→wf1o 6560 ‘cfv 6561 ωcom 7887 1st c1st 8012 2nd c2nd 8013 1oc1o 8499 inrcinr 9940 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-inr 9943 |
| This theorem is referenced by: inrresf 9956 inrresf1 9957 djuin 9958 djuun 9966 |
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