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Theorem mapval2 8865
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 (𝐴m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 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.
StepHypRef Expression
1 dff2 7093 . . . 4 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)))
21biancomi 462 . . 3 (𝑔:𝐵𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
3 elmap.1 . . . 4 𝐴 ∈ V
4 elmap.2 . . . 4 𝐵 ∈ V
53, 4elmap 8864 . . 3 (𝑔 ∈ (𝐴m 𝐵) ↔ 𝑔:𝐵𝐴)
6 elin 3959 . . . 4 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}))
7 velpw 4602 . . . . 5 (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
8 vex 3472 . . . . . 6 𝑔 ∈ V
9 fneq1 6633 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝐵𝑔 Fn 𝐵))
108, 9elab 3663 . . . . 5 (𝑔 ∈ {𝑓𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
117, 10anbi12i 626 . . . 4 ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
126, 11bitri 275 . . 3 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
132, 5, 123bitr4i 303 . 2 (𝑔 ∈ (𝐴m 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}))
1413eqriv 2723 1 (𝐴m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  cin 3942  wss 3943  𝒫 cpw 4597   × cxp 5667   Fn wfn 6531  wf 6532  (class class class)co 7404  m cmap 8819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821
This theorem is referenced by: (None)
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