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Theorem imasetpreimafvbijlemf 47388
Description: Lemma for imasetpreimafvbij 47393: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinj.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
fundcmpsurinj.h 𝐻 = (𝑝𝑃 (𝐹𝑝))
Assertion
Ref Expression
imasetpreimafvbijlemf (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃
Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)

Proof of Theorem imasetpreimafvbijlemf
StepHypRef Expression
1 fundcmpsurinj.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21uniimaelsetpreimafv 47383 . . 3 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝑝) ∈ ran 𝐹)
3 fnima 6698 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
43adantr 480 . . 3 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝐴) = ran 𝐹)
52, 4eleqtrrd 2844 . 2 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝑝) ∈ (𝐹𝐴))
6 fundcmpsurinj.h . 2 𝐻 = (𝑝𝑃 (𝐹𝑝))
75, 6fmptd 7134 1 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {csn 4626   cuni 4907  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561
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-pr 5432
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-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  imasetpreimafvbijlemf1  47391  imasetpreimafvbijlemfo  47392
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