Theorem pmex 8768

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 460	. . 3 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵)) ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓))
2	1	abbii 2803	. 2 ⊢ {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} = {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)}
3		xpexg 7695	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
4		abssexg 5327	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
5	3, 4	syl 17	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
6	2, 5	eqeltrid 2840	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  {cab 2714  Vcvv 3440   ⊆ wss 3901   × cxp 5622  Fun wfun 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 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-opab 5161  df-xp 5630  df-rel 5631

This theorem is referenced by: (None)