Theorem cfsetsnfsetfv 48096

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 6882	. . . . 5 ⊢ (𝑔 = 𝑋 → (𝑔‘𝑌) = (𝑋‘𝑌))
4	3	adantr 486	. . . 4 ⊢ ((𝑔 = 𝑋 ∧ 𝑎 ∈ 𝐴) → (𝑔‘𝑌) = (𝑋‘𝑌))
5	4	mpteq2dva 5198	. . 3 ⊢ (𝑔 = 𝑋 → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
6	5	adantl 487	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) ∧ 𝑔 = 𝑋) → (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
7		simpr 490	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝑋 ∈ 𝐺)
8		simpl 488	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → 𝐴 ∈ 𝑉)
9	8	mptexd 7228	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)) ∈ V)
10	2, 6, 7, 9	fvmptd 6999	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087  Vcvv 3451  {csn 4584   ↦ cmpt 5186  ⟶wf 6533  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  cfsetsnfsetf1  48098  cfsetsnfsetfo  48099