Theorem fabexd 7975

Description: Existence of a set of functions. In contrast to fabex 7978 or fabexg 7976, 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 7786	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
4	3	pwexd 5397	. 2 ⊢ (𝜑 → 𝒫 (𝑋 × 𝑌) ∈ V)
5		fabexd.f	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
6		fssxp 6775	. . . . . 6 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ⊆ (𝑋 × 𝑌))
7		velpw 4627	. . . . . 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 4091	. 2 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ⊆ 𝒫 (𝑋 × 𝑌))
12	4, 11	ssexd 5342	1 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  {cab 2717  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   × cxp 5698  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  fabexg  7976  f1oabexg  7980  grlimfn  47803  isgrlim  47806