![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > f10d | Structured version Visualization version GIF version |
Description: The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
f10d.f | ⊢ (𝜑 → 𝐹 = ∅) |
Ref | Expression |
---|---|
f10d | ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f10 6856 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
2 | dm0 5910 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6773 | . . . 4 ⊢ (dom ∅ = ∅ → (∅:dom ∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅:dom ∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴) |
5 | 1, 4 | mpbir 230 | . 2 ⊢ ∅:dom ∅–1-1→𝐴 |
6 | f10d.f | . . 3 ⊢ (𝜑 → 𝐹 = ∅) | |
7 | 6 | dmeqd 5895 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅) |
8 | eqidd 2725 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐴) | |
9 | 6, 7, 8 | f1eq123d 6815 | . 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴)) |
10 | 5, 9 | mpbiri 258 | 1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∅c0 4314 dom cdm 5666 –1-1→wf1 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 |
This theorem is referenced by: umgr0e 28805 usgr0e 28928 |
Copyright terms: Public domain | W3C validator |