Theorem pmex 8817

Description: The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)

Assertion

Ref	Expression
pmex	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem pmex

Step	Hyp	Ref	Expression
1		ancom 465	. . 3 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵)) ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓))
2	1	abbii 2832	. 2 ⊢ {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} = {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)}
3		xpexg 7737	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
4		abssexg 5344	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
5	3, 4	syl 18	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
6	2, 5	eqeltrid 2869	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  {cab 2743  Vcvv 3457   ⊆ wss 3907   × cxp 5650  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-opab 5168  df-xp 5658  df-rel 5659

This theorem is referenced by: (None)