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Theorem maprnin 29340
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 6004 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
2 df-f 5854 . . . . . . 7 (𝑓:𝐴𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐶))
32baibr 944 . . . . . 6 (𝑓 Fn 𝐴 → (ran 𝑓𝐶𝑓:𝐴𝐶))
41, 3syl 17 . . . . 5 (𝑓:𝐴𝐵 → (ran 𝑓𝐶𝑓:𝐴𝐶))
54pm5.32i 668 . . . 4 ((𝑓:𝐴𝐵 ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
6 maprnin.2 . . . . . 6 𝐵 ∈ V
7 maprnin.1 . . . . . 6 𝐴 ∈ V
86, 7elmap 7831 . . . . 5 (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵)
98anbi1i 730 . . . 4 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵 ∧ ran 𝑓𝐶))
10 fin 6044 . . . 4 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
115, 9, 103bitr4ri 293 . . 3 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ ran 𝑓𝐶))
1211abbii 2742 . 2 {𝑓𝑓:𝐴⟶(𝐵𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ ran 𝑓𝐶)}
136inex1 4764 . . 3 (𝐵𝐶) ∈ V
1413, 7mapval 7815 . 2 ((𝐵𝐶) ↑𝑚 𝐴) = {𝑓𝑓:𝐴⟶(𝐵𝐶)}
15 df-rab 2921 . 2 {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ran 𝑓𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ ran 𝑓𝐶)}
1612, 14, 153eqtr4i 2658 1 ((𝐵𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ran 𝑓𝐶}
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992  {cab 2612  {crab 2916  Vcvv 3191  cin 3559  wss 3560  ran crn 5080   Fn wfn 5845  wf 5846  (class class class)co 6605  𝑚 cmap 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805
This theorem is referenced by:  fpwrelmapffs  29343
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