Theorem f1omoALT 47992

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 47991 without assuming ax-un 7746. (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 6904	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
3		1oex 8503	. . . 4 ⊢ 1o ∈ V
4	3	fvconstdomi 47990	. . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o
5	2, 4	eqbrtrdi 5191	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o)
6		modom2 9276	. 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o)
7	5, 6	sylibr 233	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∃*wmo 2527  {csn 4632   class class class wbr 5152   × cxp 5680  ‘cfv 6553  1_oc1o 8486   ≼ cdom 8968

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-1o 8493  df-en 8971  df-dom 8972  df-sdom 8973

This theorem is referenced by: (None)