Theorem fnwe 7973

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⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
fnwe.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fnwe.3	⊢ (𝜑 → 𝑅 We 𝐵)
fnwe.4	⊢ (𝜑 → 𝑆 We 𝐴)
fnwe.5	⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)

Assertion

Ref	Expression
fnwe	⊢ (𝜑 → 𝑇 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.

Step	Hyp	Ref	Expression
1		fnwe.1	. 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}
2		fnwe.2	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3		fnwe.3	. 2 ⊢ (𝜑 → 𝑅 We 𝐵)
4		fnwe.4	. 2 ⊢ (𝜑 → 𝑆 We 𝐴)
5		fnwe.5	. 2 ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)
6		eqid 2738	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))}
7		eqid 2738	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩) = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)
8	1, 2, 3, 4, 5, 6, 7	fnwelem 7972	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   We wwe 5543   × cxp 5587   “ cima 5592  ⟶wf 6429  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-1st 7831  df-2nd 7832

This theorem is referenced by:  r0weon  9768