Theorem mapexOLD 8867

Description: Obsolete version of mapex 7958 as of 17-Jun-2025. (Contributed by Raph Levien, 4-Dec-2003.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
mapexOLD	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

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

Proof of Theorem mapexOLD

Step	Hyp	Ref	Expression
1		fssxp 6758	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 ⊆ (𝐴 × 𝐵))
2	1	ss2abi 4071	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
3		df-pw 4610	. . 3 ⊢ 𝒫 (𝐴 × 𝐵) = {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
4	2, 3	sseqtrri 4027	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5		xpexg 7764	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
6	5	pwexd 5385	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7		ssexg 5329	. 2 ⊢ (({𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
8	4, 6, 7	sylancr 585	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2100  {cab 2706  Vcvv 3472   ⊆ wss 3957  𝒫 cpw 4608   × cxp 5682  ⟶wf 6552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pow 5371  ax-pr 5435  ax-un 7749

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3960  df-un 3962  df-in 3964  df-ss 3974  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-opab 5217  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-fun 6558  df-fn 6559  df-f 6560

This theorem is referenced by: (None)