Theorem mapexOLD 8809

Description: Obsolete version of mapex 7917 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 6715	. . . 4 ⊢ (𝑓:𝐴⟶𝐵 → 𝑓 ⊆ (𝐴 × 𝐵))
2	1	ss2abi 4019	. . 3 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
3		df-pw 4556	. . 3 ⊢ 𝒫 (𝐴 × 𝐵) = {𝑓 ∣ 𝑓 ⊆ (𝐴 × 𝐵)}
4	2, 3	sseqtrri 3985	. 2 ⊢ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
5		xpexg 7729	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 × 𝐵) ∈ V)
6	5	pwexd 5335	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝒫 (𝐴 × 𝐵) ∈ V)
7		ssexg 5278	. 2 ⊢ (({𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ∈ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
8	4, 6, 7	sylancr 596	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  {cab 2739  Vcvv 3453   ⊆ wss 3904  𝒫 cpw 4554   × cxp 5643  ⟶wf 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-fun 6519  df-fn 6520  df-f 6521

This theorem is referenced by: (None)