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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 5354 and ax-un 7719. Variant of eufsn 47756. If existence is not needed, use mofsn 47758 or mofsn2 47759 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 5422 | . . 3 ⊢ {𝐵} ∈ V | |
4 | xpexg 7731 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐵} ∈ V) → (𝐴 × {𝐵}) ∈ V) | |
5 | 2, 3, 4 | sylancl 585 | . 2 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ V) |
6 | 1, 5 | eufsnlem 47755 | 1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∃!weu 2554 Vcvv 3466 {csn 4621 × cxp 5665 ⟶wf 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 |
This theorem is referenced by: (None) |
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