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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 6646 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
2 | dm0 5789 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6570 | . . . 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 233 | . 2 ⊢ ∅:dom ∅–1-1→𝐴 |
6 | f10d.f | . . 3 ⊢ (𝜑 → 𝐹 = ∅) | |
7 | 6 | dmeqd 5773 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅) |
8 | eqidd 2822 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐴) | |
9 | 6, 7, 8 | f1eq123d 6607 | . 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴)) |
10 | 5, 9 | mpbiri 260 | 1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∅c0 4290 dom cdm 5554 –1-1→wf1 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 |
This theorem is referenced by: umgr0e 26894 usgr0e 27017 |
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