Theorem fabexd 7867

Description: Existence of a set of functions. In contrast to fabex 7870 or fabexg 7868, 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 7684	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
4	3	pwexd 5315	. 2 ⊢ (𝜑 → 𝒫 (𝑋 × 𝑌) ∈ V)
5		fabexd.f	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
6		fssxp 6678	. . . . . 6 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ⊆ (𝑋 × 𝑌))
7		velpw 4552	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (𝑋 × 𝑌) ↔ 𝑓 ⊆ (𝑋 × 𝑌))
8	6, 7	sylibr 234	. . . . 5 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
9	5, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
10	9	ex 412	. . 3 ⊢ (𝜑 → (𝜓 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌)))
11	10	abssdv 4014	. 2 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ⊆ 𝒫 (𝑋 × 𝑌))
12	4, 11	ssexd 5260	1 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  {cab 2709  Vcvv 3436   ⊆ wss 3897  𝒫 cpw 4547   × cxp 5612  ⟶wf 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  fabexg  7868  f1oabexg  7872  grlimfn  48018  isgrlim  48021