Theorem f1omoALT 47528

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 47527 without assuming ax-un 7725. (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 6894	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
3		1oex 8476	. . . 4 ⊢ 1o ∈ V
4	3	fvconstdomi 47526	. . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o
5	2, 4	eqbrtrdi 5188	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o)
6		modom2 9245	. 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o)
7	5, 6	sylibr 233	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∃*wmo 2533  {csn 4629   class class class wbr 5149   × cxp 5675  ‘cfv 6544  1_oc1o 8459   ≼ cdom 8937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-en 8940  df-dom 8941  df-sdom 8942

This theorem is referenced by: (None)