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 6671 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
2 | dm0 5774 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6589 | . . . 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 234 | . 2 ⊢ ∅:dom ∅–1-1→𝐴 |
6 | f10d.f | . . 3 ⊢ (𝜑 → 𝐹 = ∅) | |
7 | 6 | dmeqd 5759 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅) |
8 | eqidd 2737 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐴) | |
9 | 6, 7, 8 | f1eq123d 6631 | . 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴)) |
10 | 5, 9 | mpbiri 261 | 1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∅c0 4223 dom cdm 5536 –1-1→wf1 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 |
This theorem is referenced by: umgr0e 27155 usgr0e 27278 |
Copyright terms: Public domain | W3C validator |