Theorem mapval2 8883

Description: Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Hypotheses

Ref	Expression
elmap.1	⊢ 𝐴 ∈ V
elmap.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
mapval2	⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

Distinct variable group:   𝐵,𝑓

Allowed substitution hint:   𝐴(𝑓)

Proof of Theorem mapval2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff2 7096	. . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)))
2	1	biancomi 468	. . 3 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
3		elmap.1	. . . 4 ⊢ 𝐴 ∈ V
4		elmap.2	. . . 4 ⊢ 𝐵 ∈ V
5	3, 4	elmap 8882	. . 3 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐴)
6		elin 3918	. . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}))
7		velpw 4565	. . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
8		vex 3457	. . . . . 6 ⊢ 𝑔 ∈ V
9		fneq1 6627	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵))
10	8, 9	elab 3636	. . . . 5 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
11	7, 10	anbi12i 640	. . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
12	6, 11	bitri 278	. . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
13	2, 5, 12	3bitr4i 306	. 2 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}))
14	13	eqriv 2759	1 ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560   × cxp 5657   Fn wfn 6532  ⟶wf 6533  (class class class)co 7417   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-map 8832

This theorem is used by: (None)