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Theorem maprnin 32742
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 6736 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
2 df-f 6565 . . . . . . 7 (𝑓:𝐴𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐶))
32baibr 536 . . . . . 6 (𝑓 Fn 𝐴 → (ran 𝑓𝐶𝑓:𝐴𝐶))
41, 3syl 17 . . . . 5 (𝑓:𝐴𝐵 → (ran 𝑓𝐶𝑓:𝐴𝐶))
54pm5.32i 574 . . . 4 ((𝑓:𝐴𝐵 ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
6 maprnin.2 . . . . . 6 𝐵 ∈ V
7 maprnin.1 . . . . . 6 𝐴 ∈ V
86, 7elmap 8911 . . . . 5 (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵)
98anbi1i 624 . . . 4 ((𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵 ∧ ran 𝑓𝐶))
10 fin 6788 . . . 4 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
115, 9, 103bitr4ri 304 . . 3 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶))
1211abbii 2809 . 2 {𝑓𝑓:𝐴⟶(𝐵𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
136inex1 5317 . . 3 (𝐵𝐶) ∈ V
1413, 7mapval 8878 . 2 ((𝐵𝐶) ↑m 𝐴) = {𝑓𝑓:𝐴⟶(𝐵𝐶)}
15 df-rab 3437 . 2 {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
1612, 14, 153eqtr4i 2775 1 ((𝐵𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  cin 3950  wss 3951  ran crn 5686   Fn wfn 6556  wf 6557  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868
This theorem is referenced by:  fpwrelmapffs  32745
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