Theorem pmex 6643

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 266	. . 3 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵)) ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓))
2	1	abbii 2291	. 2 ⊢ {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} = {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)}
3		xpexg 4734	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
4		abssexg 4177	. . 3 ⊢ ((𝐴 × 𝐵) ∈ V → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
5	3, 4	syl 14	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (𝑓 ⊆ (𝐴 × 𝐵) ∧ Fun 𝑓)} ∈ V)
6	2, 5	eqeltrid 2262	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2146  {cab 2161  Vcvv 2735   ⊆ wss 3127   × cxp 4618  Fun wfun 5202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-opab 4060  df-xp 4626

This theorem is referenced by: (None)