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Theorem mapval2 8293
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 6735 . . . 4 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)))
21biancomi 463 . . 3 (𝑔:𝐵𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
3 elmap.1 . . . 4 𝐴 ∈ V
4 elmap.2 . . . 4 𝐵 ∈ V
53, 4elmap 8292 . . 3 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔:𝐵𝐴)
6 elin 4096 . . . 4 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}))
7 selpw 4466 . . . . 5 (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
8 vex 3443 . . . . . 6 𝑔 ∈ V
9 fneq1 6321 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝐵𝑔 Fn 𝐵))
108, 9elab 3608 . . . . 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 276 . . 3 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
132, 5, 123bitr4i 304 . 2 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}))
1413eqriv 2794 1 (𝐴𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1525  wcel 2083  {cab 2777  Vcvv 3440  cin 3864  wss 3865  𝒫 cpw 4459   × cxp 5448   Fn wfn 6227  wf 6228  (class class class)co 7023  𝑚 cmap 8263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-map 8265
This theorem is referenced by: (None)
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