Theorem f1omoALT 48575

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 48574 without assuming ax-un 7770. (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 6922	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
3		1oex 8532	. . . 4 ⊢ 1o ∈ V
4	3	fvconstdomi 48573	. . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o
5	2, 4	eqbrtrdi 5205	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o)
6		modom2 9308	. 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o)
7	5, 6	sylibr 234	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∃*wmo 2541  {csn 4648   class class class wbr 5166   × cxp 5698  ‘cfv 6573  1_oc1o 8515   ≼ cdom 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1o 8522  df-en 9004  df-dom 9005  df-sdom 9006

This theorem is referenced by: (None)