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Theorem mapval2 6415
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 5427 . . . 4 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)))
2 ancom 262 . . . 4 ((𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
31, 2bitri 182 . . 3 (𝑔:𝐵𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
4 elmap.1 . . . 4 𝐴 ∈ V
5 elmap.2 . . . 4 𝐵 ∈ V
64, 5elmap 6414 . . 3 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔:𝐵𝐴)
7 elin 3181 . . . 4 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}))
8 selpw 3432 . . . . 5 (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
9 vex 2622 . . . . . 6 𝑔 ∈ V
10 fneq1 5088 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝐵𝑔 Fn 𝐵))
119, 10elab 2758 . . . . 5 (𝑔 ∈ {𝑓𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
128, 11anbi12i 448 . . . 4 ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
137, 12bitri 182 . . 3 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
143, 6, 133bitr4i 210 . 2 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}))
1514eqriv 2085 1 (𝐴𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wcel 1438  {cab 2074  Vcvv 2619  cin 2996  wss 2997  𝒫 cpw 3425   × cxp 4426   Fn wfn 4997  wf 4998  (class class class)co 5634  𝑚 cmap 6385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-map 6387
This theorem is referenced by: (None)
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