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Theorem wessf1orn 42227
 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 2758 . 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 42226 1 (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto→ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  𝒫 cpw 4497  {csn 4525   class class class wbr 5036   ↦ cmpt 5116   We wwe 5486  ◡ccnv 5527  ran crn 5529   ↾ cres 5530   “ cima 5531   Fn wfn 6335  –1-1-onto→wf1o 6339  ℩crio 7113 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 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  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-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114 This theorem is referenced by:  ssnnf1octb  42237  sge0resrn  43454  nnfoctbdj  43506
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