Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6030 | . . . . . 6 ⊢ (𝐹:∅–onto→𝐴 → 𝐹 Fn ∅) | |
2 | fn0 5924 | . . . . . . 7 ⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅) | |
3 | f10 6081 | . . . . . . . 8 ⊢ ∅:∅–1-1→𝐴 | |
4 | f1eq1 6009 | . . . . . . . 8 ⊢ (𝐹 = ∅ → (𝐹:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
5 | 3, 4 | mpbiri 247 | . . . . . . 7 ⊢ (𝐹 = ∅ → 𝐹:∅–1-1→𝐴) |
6 | 2, 5 | sylbi 206 | . . . . . 6 ⊢ (𝐹 Fn ∅ → 𝐹:∅–1-1→𝐴) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1→𝐴) |
8 | 7 | ancri 573 | . . . 4 ⊢ (𝐹:∅–onto→𝐴 → (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) |
9 | df-f1o 5811 | . . . 4 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹:∅–1-1→𝐴 ∧ 𝐹:∅–onto→𝐴)) | |
10 | 8, 9 | sylibr 223 | . . 3 ⊢ (𝐹:∅–onto→𝐴 → 𝐹:∅–1-1-onto→𝐴) |
11 | f1ofo 6057 | . . 3 ⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) | |
12 | 10, 11 | impbii 198 | . 2 ⊢ (𝐹:∅–onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴) |
13 | f1o00 6083 | . 2 ⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) | |
14 | 12, 13 | bitri 263 | 1 ⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∅c0 3874 Fn wfn 5799 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: fsumf1o 14301 fprodf1o 14515 0ramcl 15565 |
Copyright terms: Public domain | W3C validator |