Theorem mapval2 6656

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 5640	. . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)))
2		ancom 264	. . . 4 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
3	1, 2	bitri 183	. . 3 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
4		elmap.1	. . . 4 ⊢ 𝐴 ∈ V
5		elmap.2	. . . 4 ⊢ 𝐵 ∈ V
6	4, 5	elmap 6655	. . 3 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶𝐴)
7		elin 3310	. . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}))
8		velpw 3573	. . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
9		vex 2733	. . . . . 6 ⊢ 𝑔 ∈ V
10		fneq1 5286	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵))
11	9, 10	elab 2874	. . . . 5 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
12	8, 11	anbi12i 457	. . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
13	7, 12	bitri 183	. . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
14	3, 6, 13	3bitr4i 211	. 2 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}))
15	14	eqriv 2167	1 ⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

Syntax hints:   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {cab 2156  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  𝒫 cpw 3566   × cxp 4609   Fn wfn 5193  ⟶wf 5194  (class class class)co 5853   ↑_𝑚 cmap 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628

This theorem is referenced by: (None)