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Theorem f1omoALT 48691
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 48690 without assuming ax-un 7753. (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 8514 . . . 4 1o ∈ V
43fvconstdomi 48689 . . 3 ((𝐴 × {1o})‘𝑋) ≼ 1o
52, 4eqbrtrdi 5186 . 2 (𝜑 → (𝐹𝑋) ≼ 1o)
6 modom2 9278 . 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 1536  wcel 2105  ∃*wmo 2535  {csn 4630   class class class wbr 5147   × cxp 5686  cfv 6562  1oc1o 8497  cdom 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-1o 8504  df-en 8984  df-dom 8985  df-sdom 8986
This theorem is referenced by: (None)
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