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Mirrors > Home > MPE Home > Th. List > Mathboxes > eufsn2 | Structured version Visualization version GIF version |
Description: There is exactly one function into a singleton, assuming ax-pow 5288 and ax-un 7588. Variant of eufsn 46169. If existence is not needed, use mofsn 46171 or mofsn2 46172 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
eufsn.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
eufsn.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
eufsn2 | ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eufsn.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
2 | eufsn.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | snex 5354 | . . 3 ⊢ {𝐵} ∈ V | |
4 | xpexg 7600 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V) | |
5 | 2, 3, 4 | sylancl 586 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V) |
6 | 1, 5 | eufsnlem 46168 | 1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃!weu 2568 Vcvv 3432 {csn 4561 × cxp 5587 ⟶wf 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 |
This theorem is referenced by: (None) |
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