Theorem mapexOLD 8781

Description: Obsolete version of mapex 7893 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 6697	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 ⊆ (𝐴 × 𝐵))
2	1	ss2abi 4020	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
3		df-pw 4558	. . 3 ⊢ 𝒫 (𝐴 × 𝐵) = {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
4	2, 3	sseqtrri 3985	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5		xpexg 7705	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
6	5	pwexd 5326	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7		ssexg 5270	. 2 ⊢ (({𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
8	4, 6, 7	sylancr 588	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  {cab 2715  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   × cxp 5630  ⟶wf 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by: (None)