Theorem cfsetsnfsetfv 47035

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 6913	. . . . 5 ⊢ (𝑔 = 𝑋 → (𝑔‘𝑌) = (𝑋‘𝑌))
4	3	adantr 480	. . . 4 ⊢ ((𝑔 = 𝑋 ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) = (𝑋‘𝑌))
5	4	mpteq2dva 5251	. . 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 7251	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)) ∈ V)
10	2, 6, 7, 9	fvmptd 7030	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3481  {csn 4634   ↦ cmpt 5234  ⟶wf 6565  ‘cfv 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577

This theorem is referenced by:  cfsetsnfsetf1  47037  cfsetsnfsetfo  47038