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Theorem wessf1orn 45196
Description: Given a function 𝐹 on a well-ordered domain 𝐴 there exists a subset of 𝐴 such that 𝐹 restricted to such subset is injective and onto the range of 𝐹 (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
wessf1orn.f (𝜑𝐹 Fn 𝐴)
wessf1orn.a (𝜑𝐴𝑉)
wessf1orn.r (𝜑𝑅 We 𝐴)
Assertion
Ref Expression
wessf1orn (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑅
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem wessf1orn
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wessf1orn.f . 2 (𝜑𝐹 Fn 𝐴)
2 wessf1orn.a . 2 (𝜑𝐴𝑉)
3 wessf1orn.r . 2 (𝜑𝑅 We 𝐴)
4 eqid 2736 . 2 (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥)) = (𝑦 ∈ ran 𝐹 ↦ (𝑥 ∈ (𝐹 “ {𝑦})∀𝑧 ∈ (𝐹 “ {𝑦}) ¬ 𝑧𝑅𝑥))
51, 2, 3, 4wessf1ornlem 45195 1 (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107  wral 3060  wrex 3069  𝒫 cpw 4599  {csn 4625   class class class wbr 5142  cmpt 5224   We wwe 5635  ccnv 5683  ran crn 5685  cres 5686  cima 5687   Fn wfn 6555  1-1-ontowf1o 6559  crio 7388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389
This theorem is referenced by:  ssnnf1octb  45204  sge0resrn  46424  nnfoctbdj  46476
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