Theorem fnwe 7180

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 𝑢 𝑣 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 2610	. 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 2610	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩) = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)
8	1, 2, 3, 4, 5, 6, 7	fnwelem 7179	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   We wwe 4996   × cxp 5036   “ cima 5041  ⟶wf 5800  ‘cfv 5804  1^st c1st 7057  2^nd c2nd 7058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-1st 7059  df-2nd 7060

This theorem is referenced by:  r0weon  8718