Theorem cfsetsnfsetfv 47062

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 6860	. . . . 5 ⊢ (𝑔 = 𝑋 → (𝑔‘𝑌) = (𝑋‘𝑌))
4	3	adantr 480	. . . 4 ⊢ ((𝑔 = 𝑋 ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) = (𝑋‘𝑌))
5	4	mpteq2dva 5203	. . 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 7201	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)) ∈ V)
10	2, 6, 7, 9	fvmptd 6978	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  Vcvv 3450  {csn 4592   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  cfsetsnfsetf1  47064  cfsetsnfsetfo  47065