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Theorem f1omoALT 48793
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 48792 without assuming ax-un 7755. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
f1omoALT.1 (𝜑𝐹 = (𝐴 × {1o}))
Assertion
Ref Expression
f1omoALT (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Distinct variable groups:   𝑦,𝐹   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem f1omoALT
StepHypRef Expression
1 f1omoALT.1 . . . 4 (𝜑𝐹 = (𝐴 × {1o}))
21fveq1d 6908 . . 3 (𝜑 → (𝐹𝑋) = ((𝐴 × {1o})‘𝑋))
3 1oex 8516 . . . 4 1o ∈ V
43fvconstdomi 48791 . . 3 ((𝐴 × {1o})‘𝑋) ≼ 1o
52, 4eqbrtrdi 5182 . 2 (𝜑 → (𝐹𝑋) ≼ 1o)
6 modom2 9281 . 2 (∃*𝑦 𝑦 ∈ (𝐹𝑋) ↔ (𝐹𝑋) ≼ 1o)
75, 6sylibr 234 1 (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  ∃*wmo 2538  {csn 4626   class class class wbr 5143   × cxp 5683  cfv 6561  1oc1o 8499  cdom 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by: (None)
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