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Theorem wessf1orn 44370
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 2724 . 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 44369 1 (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2098  wral 3053  wrex 3062  𝒫 cpw 4594  {csn 4620   class class class wbr 5138  cmpt 5221   We wwe 5620  ccnv 5665  ran crn 5667  cres 5668  cima 5669   Fn wfn 6528  1-1-ontowf1o 6532  crio 7356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357
This theorem is referenced by:  ssnnf1octb  44378  sge0resrn  45605  nnfoctbdj  45657
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