Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  maprnin Structured version   Visualization version   GIF version

Theorem maprnin 31951
Description: Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
Hypotheses
Ref Expression
maprnin.1 𝐴 ∈ V
maprnin.2 𝐵 ∈ V
Assertion
Ref Expression
maprnin ((𝐵𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶}
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓

Proof of Theorem maprnin
StepHypRef Expression
1 ffn 6717 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
2 df-f 6547 . . . . . . 7 (𝑓:𝐴𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐶))
32baibr 537 . . . . . 6 (𝑓 Fn 𝐴 → (ran 𝑓𝐶𝑓:𝐴𝐶))
41, 3syl 17 . . . . 5 (𝑓:𝐴𝐵 → (ran 𝑓𝐶𝑓:𝐴𝐶))
54pm5.32i 575 . . . 4 ((𝑓:𝐴𝐵 ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
6 maprnin.2 . . . . . 6 𝐵 ∈ V
7 maprnin.1 . . . . . 6 𝐴 ∈ V
86, 7elmap 8864 . . . . 5 (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵)
98anbi1i 624 . . . 4 ((𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵 ∧ ran 𝑓𝐶))
10 fin 6771 . . . 4 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
115, 9, 103bitr4ri 303 . . 3 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶))
1211abbii 2802 . 2 {𝑓𝑓:𝐴⟶(𝐵𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
136inex1 5317 . . 3 (𝐵𝐶) ∈ V
1413, 7mapval 8831 . 2 ((𝐵𝐶) ↑m 𝐴) = {𝑓𝑓:𝐴⟶(𝐵𝐶)}
15 df-rab 3433 . 2 {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
1612, 14, 153eqtr4i 2770 1 ((𝐵𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2709  {crab 3432  Vcvv 3474  cin 3947  wss 3948  ran crn 5677   Fn wfn 6538  wf 6539  (class class class)co 7408  m cmap 8819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821
This theorem is referenced by:  fpwrelmapffs  31954
  Copyright terms: Public domain W3C validator