Theorem pmex 8404

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 463	. . 3 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵)) ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓))
2	1	abbii 2885	. 2 ⊢ {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} = {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)}
3		xpexg 7466	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
4		abssexg 5276	. . 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 2916	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113  {cab 2798  Vcvv 3491   ⊆ wss 3929   × cxp 5546  Fun wfun 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-opab 5122  df-xp 5554  df-rel 5555

This theorem is referenced by: (None)