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Theorem maprnin 31891
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 6705 . . . . . 6 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
2 df-f 6537 . . . . . . 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 8850 . . . . 5 (𝑓 ∈ (𝐵m 𝐴) ↔ 𝑓:𝐴𝐵)
98anbi1i 624 . . . 4 ((𝑓 ∈ (𝐵m 𝐴) ∧ ran 𝑓𝐶) ↔ (𝑓:𝐴𝐵 ∧ ran 𝑓𝐶))
10 fin 6759 . . . 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 5311 . . 3 (𝐵𝐶) ∈ V
1413, 7mapval 8817 . 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 3944  wss 3945  ran crn 5671   Fn wfn 6528  wf 6529  (class class class)co 7394  m cmap 8805
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 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
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 3775  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-map 8807
This theorem is referenced by:  fpwrelmapffs  31894
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