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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 9918 | . . 3 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1onn 8654 | . . . . . 6 ⊢ 1o ∈ ω | |
3 | snidg 4658 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 5709 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) | |
6 | 4, 5 | mpan 689 | . . . 4 ⊢ (𝑥 ∈ V → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
8 | fvexd 6906 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V) | |
9 | 1st2nd2 8026 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 8019 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o}) | |
11 | elsni 4641 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o) |
13 | 12 | opeq1d 4875 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈1o, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2767 | . . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈1o, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2738 | . . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (〈1o, 𝑥〉 = 𝑦 ↔ 〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉)) |
16 | eqcom 2734 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈1o, 𝑥〉) | |
17 | eqid 2727 | . . . . . . 7 ⊢ 1o = 1o | |
18 | 1oex 8490 | . . . . . . . 8 ⊢ 1o ∈ V | |
19 | vex 3473 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 18, 19 | opth 5472 | . . . . . . 7 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦))) |
21 | 17, 20 | mpbiran 708 | . . . . . 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 222 | . . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
24 | 23 | ad2antll 728 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
25 | 1, 7, 8, 24 | f1o2d 7669 | . 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V)) |
26 | 25 | mptru 1541 | 1 ⊢ inr:V–1-1-onto→({1o} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3469 {csn 4624 〈cop 4630 × cxp 5670 –1-1-onto→wf1o 6541 ‘cfv 6542 ωcom 7864 1st c1st 7985 2nd c2nd 7986 1oc1o 8473 inrcinr 9915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1st 7987 df-2nd 7988 df-1o 8480 df-inr 9918 |
This theorem is referenced by: inrresf 9931 inrresf1 9932 djuin 9933 djuun 9941 |
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