ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mapval2 GIF version

Theorem mapval2 6620
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 5610 . . . 4 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)))
2 ancom 264 . . . 4 ((𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
31, 2bitri 183 . . 3 (𝑔:𝐵𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
4 elmap.1 . . . 4 𝐴 ∈ V
5 elmap.2 . . . 4 𝐵 ∈ V
64, 5elmap 6619 . . 3 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔:𝐵𝐴)
7 elin 3290 . . . 4 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}))
8 velpw 3550 . . . . 5 (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
9 vex 2715 . . . . . 6 𝑔 ∈ V
10 fneq1 5257 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝐵𝑔 Fn 𝐵))
119, 10elab 2856 . . . . 5 (𝑔 ∈ {𝑓𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
128, 11anbi12i 456 . . . 4 ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
137, 12bitri 183 . . 3 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
143, 6, 133bitr4i 211 . 2 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}))
1514eqriv 2154 1 (𝐴𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wcel 2128  {cab 2143  Vcvv 2712  cin 3101  wss 3102  𝒫 cpw 3543   × cxp 4583   Fn wfn 5164  wf 5165  (class class class)co 5821  𝑚 cmap 6590
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-map 6592
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator