Theorem fabexd 7935

Description: Existence of a set of functions. In contrast to fabex 7937 or fabexg 7936, the condition in the class abstraction does not contain the function explicitly, but the function can be derived from it. Therefore, this theorem is also applicable for more special functions like one-to-one, onto or one-to-one onto functions. (Contributed by AV, 20-May-2025.)

Hypotheses

Ref	Expression
fabexd.f	⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
fabexd.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
fabexd.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)

Assertion

Ref	Expression
fabexd	⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

Distinct variable groups:   𝑓,𝑋   𝑓,𝑌   𝜑,𝑓

Allowed substitution hints:   𝜓(𝑓)   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem fabexd

Step	Hyp	Ref	Expression
1		fabexd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		fabexd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
3	1, 2	xpexd 7751	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
4	3	pwexd 5344	. 2 ⊢ (𝜑 → 𝒫 (𝑋 × 𝑌) ∈ V)
5		fabexd.f	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
6		fssxp 6731	. . . . . 6 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ⊆ (𝑋 × 𝑌))
7		velpw 4562	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (𝑋 × 𝑌) ↔ 𝑓 ⊆ (𝑋 × 𝑌))
8	6, 7	sylibr 237	. . . . 5 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
9	5, 8	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
10	9	ex 418	. . 3 ⊢ (𝜑 → (𝜓 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌)))
11	10	abssdv 4015	. 2 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ⊆ 𝒫 (𝑋 × 𝑌))
12	4, 11	ssexd 5289	1 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {cab 2738  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557   × cxp 5653  ⟶wf 6529

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-ext 2732  ax-sep 5251  ax-pow 5330  ax-pr 5398  ax-un 7737

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-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-fun 6535  df-fn 6536  df-f 6537

This theorem is used by:  fabexg  7936  f1oabexg  7939  grlimfn  48896  isgrlim  48899