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Mirrors > Home > MPE Home > Th. List > f1o00 | Structured version Visualization version GIF version |
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.) |
Ref | Expression |
---|---|
f1o00 | ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o4 6841 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) | |
2 | fn0 6681 | . . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅) |
5 | cnveq 5873 | . . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅) | |
6 | cnv0 6140 | . . . . . . . . . 10 ⊢ ◡∅ = ∅ | |
7 | 5, 6 | eqtrdi 2788 | . . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅) |
8 | 2, 7 | sylbi 216 | . . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅) |
9 | 8 | fneq1d 6642 | . . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
10 | 9 | biimpa 477 | . . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴) |
11 | 10 | fndmd 6654 | . . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴) |
12 | dm0 5920 | . . . . 5 ⊢ dom ∅ = ∅ | |
13 | 11, 12 | eqtr3di 2787 | . . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅) |
14 | 4, 13 | jca 512 | . . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
15 | 2 | biimpri 227 | . . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
16 | 15 | adantr 481 | . . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅) |
17 | eqid 2732 | . . . . . 6 ⊢ ∅ = ∅ | |
18 | fn0 6681 | . . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
19 | 17, 18 | mpbir 230 | . . . . 5 ⊢ ∅ Fn ∅ |
20 | 7 | fneq1d 6642 | . . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
21 | fneq2 6641 | . . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅)) | |
22 | 20, 21 | sylan9bb 510 | . . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅)) |
23 | 19, 22 | mpbiri 257 | . . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴) |
24 | 16, 23 | jca 512 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) |
25 | 14, 24 | impbii 208 | . 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
26 | 1, 25 | bitri 274 | 1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∅c0 4322 ◡ccnv 5675 dom cdm 5676 Fn wfn 6538 –1-1-onto→wf1o 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 |
This theorem is referenced by: fo00 6869 f1o0 6870 en0 9015 en0OLD 9016 en0ALT 9017 en0r 9018 infn0 9309 symgbas0 19297 derang0 34446 poimirlem28 36819 |
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