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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 5462 | . . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐹 Fn ∅) | |
2 | fn0 5357 | . . . . . . 7 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | f10 5517 | . . . . . . . 8 ⊢ ∅:∅–1-1→𝐴 | |
4 | f1eq1 5438 | . . . . . . . 8 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
5 | 3, 4 | mpbiri 168 | . . . . . . 7 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐴) |
6 | 2, 5 | sylbi 121 | . . . . . 6 ⊢ (𝐹 Fn ∅ → 𝐹:∅–1-1→𝐴) |
7 | 1, 6 | syl 14 | . . . . 5 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1→𝐴) |
8 | 7 | ancri 324 | . . . 4 ⊢ (𝐹:∅–onto→𝐴 → (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) |
9 | df-f1o 5245 | . . . 4 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) | |
10 | 8, 9 | sylibr 134 | . . 3 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1-onto→𝐴) |
11 | f1ofo 5490 | . . 3 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) | |
12 | 10, 11 | impbii 126 | . 2 ⊢ (𝐹:∅–onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴) |
13 | f1o00 5518 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
14 | 12, 13 | bitri 184 | 1 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∅c0 3437 Fn wfn 5233 –1-1→wf1 5235 –onto→wfo 5236 –1-1-onto→wf1o 5237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 |
This theorem is referenced by: enumct 7148 fsumf1o 11439 fprodf1o 11637 |
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