Theorem fnwe 8107

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 2761	. 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 2761	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩) = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)
8	1, 2, 3, 4, 5, 6, 7	fnwelem 8106	1 ⊢ (𝜑 → 𝑇 We 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587   class class class wbr 5099  {copab 5161   ↦ cmpt 5180   We wwe 5597   × cxp 5643   “ cima 5648  ⟶wf 6513  ‘cfv 6517  1^st c1st 7964  2^nd c2nd 7965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-1st 7966  df-2nd 7967

This theorem is referenced by:  r0weon  9965