Theorem fabexd 7877

Description: Existence of a set of functions. In contrast to fabex 7880 or fabexg 7878, 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 7694	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
4	3	pwexd 5322	. 2 ⊢ (𝜑 → 𝒫 (𝑋 × 𝑌) ∈ V)
5		fabexd.f	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
6		fssxp 6687	. . . . . 6 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ⊆ (𝑋 × 𝑌))
7		velpw 4557	. . . . . 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 4017	. 2 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ⊆ 𝒫 (𝑋 × 𝑌))
12	4, 11	ssexd 5267	1 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  {cab 2712  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552   × cxp 5620  ⟶wf 6486

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 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-fun 6492  df-fn 6493  df-f 6494

This theorem is referenced by:  fabexg  7878  f1oabexg  7882  grlimfn  48167  isgrlim  48170