Theorem fnwe 8075

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 2737	. 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 2737	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩) = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)
8	1, 2, 3, 4, 5, 6, 7	fnwelem 8074	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086  {copab 5148   ↦ cmpt 5167   We wwe 5576   × cxp 5622   “ cima 5627  ⟶wf 6488  ‘cfv 6492  1^st c1st 7933  2^nd c2nd 7934

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-1st 7935  df-2nd 7936

This theorem is referenced by:  r0weon  9925