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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 5229 and ax-un 7473. Variant of eufsn 45683. If existence is not needed, use mofsn 45685 or mofsn2 45686 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 5295 | . . 3 ⊢ {𝐵} ∈ V | |
4 | xpexg 7485 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V) | |
5 | 2, 3, 4 | sylancl 589 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V) |
6 | 1, 5 | eufsnlem 45682 | 1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∃!weu 2569 Vcvv 3397 {csn 4513 × cxp 5517 ⟶wf 6329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 |
This theorem is referenced by: (None) |
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