Theorem fabexd 7918

Description: Existence of a set of functions. In contrast to fabex 7920 or fabexg 7919, 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 7734	. . 3 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ V)
4	3	pwexd 5336	. 2 ⊢ (𝜑 → 𝒫 (𝑋 × 𝑌) ∈ V)
5		fabexd.f	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)
6		fssxp 6719	. . . . . 6 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ⊆ (𝑋 × 𝑌))
7		velpw 4560	. . . . . 6 ⊢ (𝑓 ∈ 𝒫 (𝑋 × 𝑌) ↔ 𝑓 ⊆ (𝑋 × 𝑌))
8	6, 7	sylibr 236	. . . . 5 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
9	5, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑓 ∈ 𝒫 (𝑋 × 𝑌))
10	9	ex 416	. . 3 ⊢ (𝜑 → (𝜓 → 𝑓 ∈ 𝒫 (𝑋 × 𝑌)))
11	10	abssdv 4020	. 2 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ⊆ 𝒫 (𝑋 × 𝑌))
12	4, 11	ssexd 5280	1 ⊢ (𝜑 → {𝑓 ∣ 𝜓} ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  {cab 2740  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555   × cxp 5645  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-fun 6523  df-fn 6524  df-f 6525

This theorem is referenced by:  fabexg  7919  f1oabexg  7922  grlimfn  48598  isgrlim  48601