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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 6759 | . . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐹 Fn ∅) | |
2 | fn0 6633 | . . . . . . 7 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | f10 6818 | . . . . . . . 8 ⊢ ∅:∅–1-1→𝐴 | |
4 | f1eq1 6734 | . . . . . . . 8 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
5 | 3, 4 | mpbiri 258 | . . . . . . 7 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐴) |
6 | 2, 5 | sylbi 216 | . . . . . 6 ⊢ (𝐹 Fn ∅ → 𝐹:∅–1-1→𝐴) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1→𝐴) |
8 | 7 | ancri 551 | . . . 4 ⊢ (𝐹:∅–onto→𝐴 → (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) |
9 | df-f1o 6504 | . . . 4 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) | |
10 | 8, 9 | sylibr 233 | . . 3 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1-onto→𝐴) |
11 | f1ofo 6792 | . . 3 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) | |
12 | 10, 11 | impbii 208 | . 2 ⊢ (𝐹:∅–onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴) |
13 | f1o00 6820 | . 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 205 ∧ wa 397 = wceq 1542 ∅c0 4283 Fn wfn 6492 –1-1→wf1 6494 –onto→wfo 6495 –1-1-onto→wf1o 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: fsumf1o 15613 fprodf1o 15834 0ramcl 16900 fullthinc 47152 |
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