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Theorem f1omoALT 47799
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 47798 without assuming ax-un 7722. (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 6887 . . 3 (𝜑 → (𝐹𝑋) = ((𝐴 × {1o})‘𝑋))
3 1oex 8477 . . . 4 1o ∈ V
43fvconstdomi 47797 . . 3 ((𝐴 × {1o})‘𝑋) ≼ 1o
52, 4eqbrtrdi 5180 . 2 (𝜑 → (𝐹𝑋) ≼ 1o)
6 modom2 9247 . 2 (∃*𝑦 𝑦 ∈ (𝐹𝑋) ↔ (𝐹𝑋) ≼ 1o)
75, 6sylibr 233 1 (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  ∃*wmo 2526  {csn 4623   class class class wbr 5141   × cxp 5667  cfv 6537  1oc1o 8460  cdom 8939
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-pow 5356  ax-pr 5420  ax-un 7722
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-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  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-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8467  df-en 8942  df-dom 8943  df-sdom 8944
This theorem is referenced by: (None)
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