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Theorem eufsnlem 47781
Description: There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 47782 assuming ax-rep 5278, or eufsn2 47783 assuming ax-pow 5356 and ax-un 7722. (Contributed by Zhi Wang, 19-Sep-2024.)
Hypotheses
Ref Expression
eufsn.1 (𝜑𝐵𝑊)
eufsnlem.2 (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)
Assertion
Ref Expression
eufsnlem (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem eufsnlem
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eufsnlem.2 . . 3 (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)
2 eufsn.1 . . . . 5 (𝜑𝐵𝑊)
3 fconst2g 7200 . . . . 5 (𝐵𝑊 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
42, 3syl 17 . . . 4 (𝜑 → (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
54alrimiv 1922 . . 3 (𝜑 → ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵})))
6 eqeq2 2738 . . . . 5 (𝑔 = (𝐴 × {𝐵}) → (𝑓 = 𝑔𝑓 = (𝐴 × {𝐵})))
76bibi2d 342 . . . 4 (𝑔 = (𝐴 × {𝐵}) → ((𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ (𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
87albidv 1915 . . 3 (𝑔 = (𝐴 × {𝐵}) → (∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) ↔ ∀𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = (𝐴 × {𝐵}))))
91, 5, 8spcedv 3582 . 2 (𝜑 → ∃𝑔𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔))
10 eu6im 2563 . 2 (∃𝑔𝑓(𝑓:𝐴⟶{𝐵} ↔ 𝑓 = 𝑔) → ∃!𝑓 𝑓:𝐴⟶{𝐵})
119, 10syl 17 1 (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2556  {csn 4623   × cxp 5667  wf 6533
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by:  eufsn  47782  eufsn2  47783
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