Theorem f1omoALT 50117

Description: There is at most one element in the function value of a constant function whose output is 1_o. (An artifact of our function value definition.) Use f1omo 50115 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	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omoALT	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐹   𝑦,𝑋

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem f1omoALT

Step	Hyp	Ref	Expression
1		f1omoALT.1	. . . 4 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
2	1	fveq1d 6892	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
3		1oex 8493	. . . 4 ⊢ 1o ∈ V
4	3	fvconstdomi 50114	. . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o
5	2, 4	eqbrtrdi 5143	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o)
6		modom2 9250	. 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o)
7	5, 6	sylibr 237	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∃*wmo 2562  {csn 4584   class class class wbr 5102   × cxp 5649  ‘cfv 6543  1_oc1o 8476   ≼ cdom 8978

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8483  df-en 8981  df-dom 8982  df-sdom 8983

This theorem is used by: (None)