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Mirrors > Home > MPE Home > Th. List > djulf1o | Structured version Visualization version GIF version |
Description: The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
Ref | Expression |
---|---|
djulf1o | ⊢ inl:V–1-1-onto→({∅} × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 9324 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | 0ex 5204 | . . . . . 6 ⊢ ∅ ∈ V | |
3 | 2 | snid 4594 | . . . . 5 ⊢ ∅ ∈ {∅} |
4 | opelxpi 5585 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) | |
5 | 3, 4 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ V → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
6 | 5 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ V) → 〈∅, 𝑥〉 ∈ ({∅} × V)) |
7 | fvexd 6678 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ({∅} × V)) → (2nd ‘𝑦) ∈ V) | |
8 | 1st2nd2 7721 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | |
9 | xp1st 7714 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) ∈ {∅}) | |
10 | elsni 4577 | . . . . . . . . . 10 ⊢ ((1st ‘𝑦) ∈ {∅} → (1st ‘𝑦) = ∅) | |
11 | 9, 10 | syl 17 | . . . . . . . . 9 ⊢ (𝑦 ∈ ({∅} × V) → (1st ‘𝑦) = ∅) |
12 | 11 | opeq1d 4802 | . . . . . . . 8 ⊢ (𝑦 ∈ ({∅} × V) → 〈(1st ‘𝑦), (2nd ‘𝑦)〉 = 〈∅, (2nd ‘𝑦)〉) |
13 | 8, 12 | eqtrd 2855 | . . . . . . 7 ⊢ (𝑦 ∈ ({∅} × V) → 𝑦 = 〈∅, (2nd ‘𝑦)〉) |
14 | 13 | eqeq2d 2831 | . . . . . 6 ⊢ (𝑦 ∈ ({∅} × V) → (〈∅, 𝑥〉 = 𝑦 ↔ 〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉)) |
15 | eqcom 2827 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 𝑦 ↔ 𝑦 = 〈∅, 𝑥〉) | |
16 | eqid 2820 | . . . . . . 7 ⊢ ∅ = ∅ | |
17 | vex 3494 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
18 | 2, 17 | opth 5361 | . . . . . . 7 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ (∅ = ∅ ∧ 𝑥 = (2nd ‘𝑦))) |
19 | 16, 18 | mpbiran 707 | . . . . . 6 ⊢ (〈∅, 𝑥〉 = 〈∅, (2nd ‘𝑦)〉 ↔ 𝑥 = (2nd ‘𝑦)) |
20 | 14, 15, 19 | 3bitr3g 315 | . . . . 5 ⊢ (𝑦 ∈ ({∅} × V) → (𝑦 = 〈∅, 𝑥〉 ↔ 𝑥 = (2nd ‘𝑦))) |
21 | 20 | bicomd 225 | . . . 4 ⊢ (𝑦 ∈ ({∅} × V) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
22 | 21 | ad2antll 727 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ ({∅} × V))) → (𝑥 = (2nd ‘𝑦) ↔ 𝑦 = 〈∅, 𝑥〉)) |
23 | 1, 6, 7, 22 | f1o2d 7392 | . 2 ⊢ (⊤ → inl:V–1-1-onto→({∅} × V)) |
24 | 23 | mptru 1543 | 1 ⊢ inl:V–1-1-onto→({∅} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {csn 4560 〈cop 4566 × cxp 5546 –1-1-onto→wf1o 6347 ‘cfv 6348 1st c1st 7680 2nd c2nd 7681 inlcinl 9321 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-1st 7682 df-2nd 7683 df-inl 9324 |
This theorem is referenced by: inlresf 9336 inlresf1 9337 djuin 9340 djuun 9348 |
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