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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 9335 | . . 3 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
2 | 1onn 8268 | . . . . . 6 ⊢ 1o ∈ ω | |
3 | snidg 4602 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ {1o}) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 5595 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ V → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
7 | 6 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈1o, 𝑥〉 ∈ ({1o} × V)) |
8 | fvexd 6688 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({1o} × V)) → (2nd ‘𝑦) ∈ V) | |
9 | 1st2nd2 7731 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
10 | xp1st 7724 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) ∈ {1o}) | |
11 | elsni 4587 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {1o} → (1st ‘𝑦) = 1o) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({1o} × V) → (1st ‘𝑦) = 1o) |
13 | 12 | opeq1d 4812 | . . . . . . . 8 ⊢ (𝑦 ∈ ({1o} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈1o, (2nd ‘𝑦)〉) |
14 | 9, 13 | eqtrd 2859 | . . . . . . 7 ⊢ (𝑦 ∈ ({1o} × V) → 𝑦 = 〈1o, (2nd ‘𝑦)〉) |
15 | 14 | eqeq2d 2835 | . . . . . 6 ⊢ (𝑦 ∈ ({1o} × V) → (〈1o, 𝑥〉 = 𝑦 ↔ 〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉)) |
16 | eqcom 2831 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈1o, 𝑥〉) | |
17 | eqid 2824 | . . . . . . 7 ⊢ 1o = 1o | |
18 | 1oex 8113 | . . . . . . . 8 ⊢ 1o ∈ V | |
19 | vex 3500 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
20 | 18, 19 | opth 5371 | . . . . . . 7 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ (1o = 1o ∧ 𝑥 = (2nd ‘𝑦))) |
21 | 17, 20 | mpbiran 707 | . . . . . 6 ⊢ (〈1o, 𝑥〉 = 〈1o, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
22 | 15, 16, 21 | 3bitr3g 315 | . . . . 5 ⊢ (𝑦 ∈ ({1o} × V) → (𝑦 = 〈1o, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
23 | 22 | bicomd 225 | . . . 4 ⊢ (𝑦 ∈ ({1o} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
24 | 23 | ad2antll 727 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({1o} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈1o, 𝑥〉)) |
25 | 1, 7, 8, 24 | f1o2d 7402 | . 2 ⊢ (⊤ → inr:V–1-1-onto→({1o} × V)) |
26 | 25 | mptru 1543 | 1 ⊢ inr:V–1-1-onto→({1o} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 × cxp 5556 –1-1-onto→wf1o 6357 ‘cfv 6358 ωcom 7583 1st c1st 7690 2nd c2nd 7691 1oc1o 8098 inrcinr 9332 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-1st 7692 df-2nd 7693 df-1o 8105 df-inr 9335 |
This theorem is referenced by: inrresf 9348 inrresf1 9349 djuin 9350 djuun 9358 |
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