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Theorem mapval2 8039
 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 6514 . . . 4 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)))
2 ancom 452 . . . 4 ((𝑔 Fn 𝐵𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
31, 2bitri 264 . . 3 (𝑔:𝐵𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
4 elmap.1 . . . 4 𝐴 ∈ V
5 elmap.2 . . . 4 𝐵 ∈ V
64, 5elmap 8038 . . 3 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔:𝐵𝐴)
7 elin 3947 . . . 4 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}))
8 selpw 4304 . . . . 5 (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
9 vex 3354 . . . . . 6 𝑔 ∈ V
10 fneq1 6119 . . . . . 6 (𝑓 = 𝑔 → (𝑓 Fn 𝐵𝑔 Fn 𝐵))
119, 10elab 3501 . . . . 5 (𝑔 ∈ {𝑓𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
128, 11anbi12i 612 . . . 4 ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
137, 12bitri 264 . . 3 (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
143, 6, 133bitr4i 292 . 2 (𝑔 ∈ (𝐴𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵}))
1514eqriv 2768 1 (𝐴𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓𝑓 Fn 𝐵})
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4297   × cxp 5247   Fn wfn 6026  ⟶wf 6027  (class class class)co 6793   ↑𝑚 cmap 8009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011 This theorem is referenced by: (None)
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