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Mirrors > Home > MPE Home > Th. List > fo00 | Structured version Visualization version 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 6822 | . . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐹 Fn ∅) | |
2 | fn0 6699 | . . . . . . 7 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | f10 6881 | . . . . . . . 8 ⊢ ∅:∅–1-1→𝐴 | |
4 | f1eq1 6799 | . . . . . . . 8 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
5 | 3, 4 | mpbiri 258 | . . . . . . 7 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐴) |
6 | 2, 5 | sylbi 217 | . . . . . 6 ⊢ (𝐹 Fn ∅ → 𝐹:∅–1-1→𝐴) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1→𝐴) |
8 | 7 | ancri 549 | . . . 4 ⊢ (𝐹:∅–onto→𝐴 → (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) |
9 | df-f1o 6569 | . . . 4 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) | |
10 | 8, 9 | sylibr 234 | . . 3 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1-onto→𝐴) |
11 | f1ofo 6855 | . . 3 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) | |
12 | 10, 11 | impbii 209 | . 2 ⊢ (𝐹:∅–onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴) |
13 | f1o00 6883 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
14 | 12, 13 | bitri 275 | 1 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∅c0 4338 Fn wfn 6557 –1-1→wf1 6559 –onto→wfo 6560 –1-1-onto→wf1o 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 |
This theorem is referenced by: fsumf1o 15755 fprodf1o 15978 0ramcl 17056 fullthinc 48845 |
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