Theorem cfsetsnfsetfv 47527

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 6833	. . . . 5 ⊢ (𝑔 = 𝑋 → (𝑔‘𝑌) = (𝑋‘𝑌))
4	3	adantr 481	. . . 4 ⊢ ((𝑔 = 𝑋 ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) = (𝑋‘𝑌))
5	4	mpteq2dva 5172	. . 3 ⊢ (𝑔 = 𝑋 → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
6	5	adantl 482	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) ∧ 𝑔 = 𝑋) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
7		simpr 485	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝑋 ∈ 𝐺)
8		simpl 483	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝐴 ∈ 𝑉)
9	8	mptexd 7175	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)) ∈ V)
10	2, 6, 7, 9	fvmptd 6950	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064  Vcvv 3432  {csn 4562   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  cfsetsnfsetf1  47529  cfsetsnfsetfo  47530