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Theorem imasetpreimafvbij 46008
Description: The mapping 𝐻 is a bijective function betwen 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 46006 . . 3 (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
43adantr 482 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1→(𝐹𝐴))
51, 2imasetpreimafvbijlemfo 46007 . 2 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
6 df-f1o 6546 . 2 (𝐻:𝑃1-1-onto→(𝐹𝐴) ↔ (𝐻:𝑃1-1→(𝐹𝐴) ∧ 𝐻:𝑃onto→(𝐹𝐴)))
74, 5, 6sylanbrc 584 1 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1-onto→(𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  {csn 4626   cuni 4906  cmpt 5229  ccnv 5673  cima 5677   Fn wfn 6534  1-1wf1 6536  ontowfo 6537  1-1-ontowf1o 6538  cfv 6539
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-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
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-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547
This theorem is referenced by:  fundcmpsurbijinjpreimafv  46009
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