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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 6640 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
2 | dm0 5783 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6564 | . . . 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 232 | . 2 ⊢ ∅:dom ∅–1-1→𝐴 |
6 | f10d.f | . . 3 ⊢ (𝜑 → 𝐹 = ∅) | |
7 | 6 | dmeqd 5767 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅) |
8 | eqidd 2819 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐴) | |
9 | 6, 7, 8 | f1eq123d 6601 | . 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴)) |
10 | 5, 9 | mpbiri 259 | 1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∅c0 4288 dom cdm 5548 –1-1→wf1 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 |
This theorem is referenced by: umgr0e 26822 usgr0e 26945 |
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