Theorem cfsetsnfsetfv 47094

Description: The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-Sep-2024.)

Hypotheses

Ref	Expression
cfsetsnfsetfv.f	⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}
cfsetsnfsetfv.g	⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}
cfsetsnfsetfv.h	⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))

Assertion

Ref	Expression
cfsetsnfsetfv	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

Distinct variable groups:   𝐴,𝑎,𝑔   𝑔,𝐺   𝑔,𝑉   𝑋,𝑎,𝑔   𝑔,𝑌

Allowed substitution hints:   𝐴(𝑥,𝑧,𝑓,𝑏)   𝐵(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝐹(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝐺(𝑥,𝑧,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑧,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑧,𝑓,𝑎,𝑏)   𝑋(𝑥,𝑧,𝑓,𝑏)   𝑌(𝑥,𝑧,𝑓,𝑎,𝑏)

Proof of Theorem cfsetsnfsetfv

Step	Hyp	Ref	Expression
1		cfsetsnfsetfv.h	. . 3 ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))))
3		fveq1 6821	. . . . 5 ⊢ (𝑔 = 𝑋 → (𝑔‘𝑌) = (𝑋‘𝑌))
4	3	adantr 480	. . . 4 ⊢ ((𝑔 = 𝑋 ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) = (𝑋‘𝑌))
5	4	mpteq2dva 5184	. . 3 ⊢ (𝑔 = 𝑋 → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
6	5	adantl 481	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) ∧ 𝑔 = 𝑋) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
7		simpr 484	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝑋 ∈ 𝐺)
8		simpl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝐴 ∈ 𝑉)
9	8	mptexd 7158	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)) ∈ V)
10	2, 6, 7, 9	fvmptd 6936	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436  {csn 4576   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by:  cfsetsnfsetf1  47096  cfsetsnfsetfo  47097