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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 6425 | . . 3 ⊢ ∅:∅–1-1→𝐴 | |
2 | dm0 5586 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6349 | . . . 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 223 | . 2 ⊢ ∅:dom ∅–1-1→𝐴 |
6 | f10d.f | . . 3 ⊢ (𝜑 → 𝐹 = ∅) | |
7 | 6 | dmeqd 5573 | . . 3 ⊢ (𝜑 → dom 𝐹 = dom ∅) |
8 | eqidd 2779 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐴) | |
9 | 6, 7, 8 | f1eq123d 6386 | . 2 ⊢ (𝜑 → (𝐹:dom 𝐹–1-1→𝐴 ↔ ∅:dom ∅–1-1→𝐴)) |
10 | 5, 9 | mpbiri 250 | 1 ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∅c0 4141 dom cdm 5357 –1-1→wf1 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 |
This theorem is referenced by: umgr0e 26462 usgr0e 26587 |
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