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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 6647 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) | |
2 | fn0 6487 | . . . . . 6 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | 2 | biimpi 219 | . . . . 5 ⊢ (𝐹 Fn ∅ → 𝐹 = ∅) |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐹 = ∅) |
5 | cnveq 5727 | . . . . . . . . . 10 ⊢ (𝐹 = ∅ → ◡𝐹 = ◡∅) | |
6 | cnv0 5984 | . . . . . . . . . 10 ⊢ ◡∅ = ∅ | |
7 | 5, 6 | eqtrdi 2787 | . . . . . . . . 9 ⊢ (𝐹 = ∅ → ◡𝐹 = ∅) |
8 | 2, 7 | sylbi 220 | . . . . . . . 8 ⊢ (𝐹 Fn ∅ → ◡𝐹 = ∅) |
9 | 8 | fneq1d 6450 | . . . . . . 7 ⊢ (𝐹 Fn ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
10 | 9 | biimpa 480 | . . . . . 6 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → ∅ Fn 𝐴) |
11 | 10 | fndmd 6461 | . . . . 5 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → dom ∅ = 𝐴) |
12 | dm0 5774 | . . . . 5 ⊢ dom ∅ = ∅ | |
13 | 11, 12 | eqtr3di 2786 | . . . 4 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → 𝐴 = ∅) |
14 | 4, 13 | jca 515 | . . 3 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅)) |
15 | 2 | biimpri 231 | . . . . 5 ⊢ (𝐹 = ∅ → 𝐹 Fn ∅) |
16 | 15 | adantr 484 | . . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅) |
17 | eqid 2736 | . . . . . 6 ⊢ ∅ = ∅ | |
18 | fn0 6487 | . . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅) | |
19 | 17, 18 | mpbir 234 | . . . . 5 ⊢ ∅ Fn ∅ |
20 | 7 | fneq1d 6450 | . . . . . 6 ⊢ (𝐹 = ∅ → (◡𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴)) |
21 | fneq2 6449 | . . . . . 6 ⊢ (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅)) | |
22 | 20, 21 | sylan9bb 513 | . . . . 5 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (◡𝐹 Fn 𝐴 ↔ ∅ Fn ∅)) |
23 | 19, 22 | mpbiri 261 | . . . 4 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → ◡𝐹 Fn 𝐴) |
24 | 16, 23 | jca 515 | . . 3 ⊢ ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴)) |
25 | 14, 24 | impbii 212 | . 2 ⊢ ((𝐹 Fn ∅ ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
26 | 1, 25 | bitri 278 | 1 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∅c0 4223 ◡ccnv 5535 dom cdm 5536 Fn wfn 6353 –1-1-onto→wf1o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 |
This theorem is referenced by: fo00 6674 f1o0 6675 en0 8669 en0OLD 8670 en0ALT 8671 symgbas0 18735 derang0 32798 poimirlem28 35491 |
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