Theorem mapexOLD 8762

Description: Obsolete version of mapex 7877 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 6683	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 ⊆ (𝐴 × 𝐵))
2	1	ss2abi 4015	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
3		df-pw 4551	. . 3 ⊢ 𝒫 (𝐴 × 𝐵) = {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
4	2, 3	sseqtrri 3980	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5		xpexg 7689	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
6	5	pwexd 5319	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7		ssexg 5263	. 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 2113  {cab 2711  Vcvv 3437   ⊆ wss 3898  𝒫 cpw 4549   × cxp 5617  ⟶wf 6482

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 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by: (None)