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Theorem f1omoALT 45759
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 45758 without assuming ax-un 7491. (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 6688 . . 3 (𝜑 → (𝐹𝑋) = ((𝐴 × {1o})‘𝑋))
3 1oex 8156 . . . 4 1o ∈ V
43fvconstdomi 45757 . . 3 ((𝐴 × {1o})‘𝑋) ≼ 1o
52, 4eqbrtrdi 5079 . 2 (𝜑 → (𝐹𝑋) ≼ 1o)
6 modom2 8811 . 2 (∃*𝑦 𝑦 ∈ (𝐹𝑋) ↔ (𝐹𝑋) ≼ 1o)
75, 6sylibr 237 1 (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  ∃*wmo 2539  {csn 4526   class class class wbr 5040   × cxp 5533  cfv 6349  1oc1o 8136  cdom 8565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7612  df-1o 8143  df-er 8332  df-en 8568  df-dom 8569  df-sdom 8570
This theorem is referenced by: (None)
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