Theorem f1omoALT 49477

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 49475 without assuming ax-un 7713. (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 6864	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
3		1oex 8441	. . . 4 ⊢ 1o ∈ V
4	3	fvconstdomi 49474	. . 3 ⊢ ((𝐴 × {1o})‘𝑋) ≼ 1o
5	2, 4	eqbrtrdi 5136	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≼ 1o)
6		modom2 9190	. 2 ⊢ (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ (𝐹‘𝑋) ≼ 1o)
7	5, 6	sylibr 236	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∃*wmo 2563  {csn 4579   class class class wbr 5097   × cxp 5641  ‘cfv 6516  1_oc1o 8424   ≼ cdom 8919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-1o 8431  df-en 8922  df-dom 8923  df-sdom 8924

This theorem is referenced by: (None)