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Theorem maprnin 31702
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 6672 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
2 df-f 6504 . . . . . . 7 (𝑓:𝐴𝐶 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐶))
32baibr 538 . . . . . 6 (𝑓 Fn 𝐴 → (ran 𝑓𝐶𝑓:𝐴𝐶))
41, 3syl 17 . . . . 5 (𝑓:𝐴𝐵 → (ran 𝑓𝐶𝑓:𝐴𝐶))
54pm5.32i 576 . . . 4 ((𝑓:𝐴𝐵 ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
6 maprnin.2 . . . . . 6 𝐵 ∈ V
7 maprnin.1 . . . . . 6 𝐴 ∈ V
86, 7elmap 8815 . . . . 5 (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵)
98anbi1i 625 . . . 4 ((𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵 ∧ ran 𝑓𝐶))
10 fin 6726 . . . 4 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓:𝐴𝐵𝑓:𝐴𝐶))
115, 9, 103bitr4ri 304 . . 3 (𝑓:𝐴⟶(𝐵𝐶) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶))
1211abbii 2803 . 2 {𝑓𝑓:𝐴⟶(𝐵𝐶)} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
136inex1 5278 . . 3 (𝐵𝐶) ∈ V
1413, 7mapval 8783 . 2 ((𝐵𝐶) ↑m 𝐴) = {𝑓𝑓:𝐴⟶(𝐵𝐶)}
15 df-rab 3407 . 2 {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶} = {𝑓 ∣ (𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶)}
1612, 14, 153eqtr4i 2771 1 ((𝐵𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  {crab 3406  Vcvv 3447  cin 3913  wss 3914  ran crn 5638   Fn wfn 6495  wf 6496  (class class class)co 7361  m cmap 8771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773
This theorem is referenced by:  fpwrelmapffs  31705
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