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Mirrors > Home > MPE Home > Th. List > Mathboxes > eufsnlem | Structured version Visualization version GIF version |
Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 45785 assuming ax-rep 5164, or eufsn2 45786 assuming ax-pow 5243 and ax-un 7501. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
eufsn.1 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
eufsnlem.2 | ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉) |
Ref | Expression |
---|---|
eufsnlem | ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eufsnlem.2 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉) | |
2 | eufsn.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | fconst2g 6996 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))) |
5 | 4 | alrimiv 1935 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))) |
6 | eqeq2 2748 | . . . . 5 ⊢ (𝑔 = (𝐴 × {𝐵}) → (𝑓 = 𝑔 ↔ 𝑓 = (𝐴 × {𝐵}))) | |
7 | 6 | bibi2d 346 | . . . 4 ⊢ (𝑔 = (𝐴 × {𝐵}) → ((𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))) |
8 | 7 | albidv 1928 | . . 3 ⊢ (𝑔 = (𝐴 × {𝐵}) → (∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))) |
9 | 1, 5, 8 | spcedv 3503 | . 2 ⊢ (𝜑 → ∃𝑔∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔)) |
10 | eu6im 2574 | . 2 ⊢ (∃𝑔∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) → ∃!𝑓 𝑓:𝐴⟶{𝐵}) | |
11 | 9, 10 | syl 17 | 1 ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∃!weu 2567 {csn 4527 × cxp 5534 ⟶wf 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 |
This theorem is referenced by: eufsn 45785 eufsn2 45786 |
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