Theorem mapexOLD 8871

Description: Obsolete version of mapex 7962 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 6764	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 ⊆ (𝐴 × 𝐵))
2	1	ss2abi 4077	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
3		df-pw 4607	. . 3 ⊢ 𝒫 (𝐴 × 𝐵) = {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
4	2, 3	sseqtrri 4033	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5		xpexg 7769	. . 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 587	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  {cab 2712  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605   × cxp 5687  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by: (None)