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Theorem fnwe 7817
Description: A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
fnwe.1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
fnwe.2 (𝜑𝐹:𝐴𝐵)
fnwe.3 (𝜑𝑅 We 𝐵)
fnwe.4 (𝜑𝑆 We 𝐴)
fnwe.5 (𝜑 → (𝐹𝑤) ∈ V)
Assertion
Ref Expression
fnwe (𝜑𝑇 We 𝐴)
Distinct variable groups:   𝑥,𝑤,𝑦,𝐴   𝑤,𝐵,𝑥,𝑦   𝜑,𝑤,𝑥   𝑤,𝐹,𝑥,𝑦   𝑤,𝑅,𝑥,𝑦   𝑤,𝑆,𝑥,𝑦   𝑤,𝑇
Allowed substitution hints:   𝜑(𝑦)   𝑇(𝑥,𝑦)

Proof of Theorem fnwe
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnwe.1 . 2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐴) ∧ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑆𝑦)))}
2 fnwe.2 . 2 (𝜑𝐹:𝐴𝐵)
3 fnwe.3 . 2 (𝜑𝑅 We 𝐵)
4 fnwe.4 . 2 (𝜑𝑆 We 𝐴)
5 fnwe.5 . 2 (𝜑 → (𝐹𝑤) ∈ V)
6 eqid 2821 . 2 {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st𝑢)𝑅(1st𝑣) ∨ ((1st𝑢) = (1st𝑣) ∧ (2nd𝑢)𝑆(2nd𝑣))))}
7 eqid 2821 . 2 (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩) = (𝑧𝐴 ↦ ⟨(𝐹𝑧), 𝑧⟩)
81, 2, 3, 4, 5, 6, 7fnwelem 7816 1 (𝜑𝑇 We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  Vcvv 3495  cop 4565   class class class wbr 5058  {copab 5120  cmpt 5138   We wwe 5507   × cxp 5547  cima 5552  wf 6345  cfv 6349  1st c1st 7678  2nd c2nd 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-int 4870  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-1st 7680  df-2nd 7681
This theorem is referenced by:  r0weon  9427
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