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Theorem imasetpreimafvbijlemf 44286
 Description: Lemma for imasetpreimafvbij 44291: 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 44281 . . 3 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝑝) ∈ ran 𝐹)
3 fnima 6461 . . . 4 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
43adantr 484 . . 3 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝐴) = ran 𝐹)
52, 4eleqtrrd 2855 . 2 ((𝐹 Fn 𝐴𝑝𝑃) → (𝐹𝑝) ∈ (𝐹𝐴))
6 fundcmpsurinj.h . 2 𝐻 = (𝑝𝑃 (𝐹𝑝))
75, 6fmptd 6869 1 (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071  {csn 4522  ∪ cuni 4798   ↦ cmpt 5112  ◡ccnv 5523  ran crn 5525   “ cima 5527   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343 This theorem is referenced by:  imasetpreimafvbijlemf1  44289  imasetpreimafvbijlemfo  44290
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