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Mirrors > Home > ILE Home > Th. List > fo00 | GIF version |
Description: Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.) |
Ref | Expression |
---|---|
fo00 | ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofn 5406 | . . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐹 Fn ∅) | |
2 | fn0 5301 | . . . . . . 7 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | f10 5460 | . . . . . . . 8 ⊢ ∅:∅–1-1→𝐴 | |
4 | f1eq1 5382 | . . . . . . . 8 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
5 | 3, 4 | mpbiri 167 | . . . . . . 7 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐴) |
6 | 2, 5 | sylbi 120 | . . . . . 6 ⊢ (𝐹 Fn ∅ → 𝐹:∅–1-1→𝐴) |
7 | 1, 6 | syl 14 | . . . . 5 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1→𝐴) |
8 | 7 | ancri 322 | . . . 4 ⊢ (𝐹:∅–onto→𝐴 → (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) |
9 | df-f1o 5189 | . . . 4 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) | |
10 | 8, 9 | sylibr 133 | . . 3 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1-onto→𝐴) |
11 | f1ofo 5433 | . . 3 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) | |
12 | 10, 11 | impbii 125 | . 2 ⊢ (𝐹:∅–onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴) |
13 | f1o00 5461 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
14 | 12, 13 | bitri 183 | 1 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∅c0 3404 Fn wfn 5177 –1-1→wf1 5179 –onto→wfo 5180 –1-1-onto→wf1o 5181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 |
This theorem is referenced by: enumct 7071 fsumf1o 11317 fprodf1o 11515 |
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