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Theorem imasetpreimafvbij 47331
Description: The mapping 𝐻 is a bijective function between the set 𝑃 of all preimages of values of function 𝐹 and the range of 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
imasetpreimafvbij ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1-onto→(𝐹𝐴))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑉,𝑝
Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑉(𝑥,𝑧)

Proof of Theorem imasetpreimafvbij
StepHypRef Expression
1 fundcmpsurinj.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
2 fundcmpsurinj.h . . . 4 𝐻 = (𝑝𝑃 (𝐹𝑝))
31, 2imasetpreimafvbijlemf1 47329 . . 3 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
43adantr 480 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1→(𝐹𝐴))
51, 2imasetpreimafvbijlemfo 47330 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
6 df-f1o 6570 . 2 (𝐻:𝑃1-1-onto→(𝐹𝐴) ↔ (𝐻:𝑃1-1→(𝐹𝐴) ∧ 𝐻:𝑃onto→(𝐹𝐴)))
74, 5, 6sylanbrc 583 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1-onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  cima 5692   Fn wfn 6558  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  fundcmpsurbijinjpreimafv  47332
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