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Theorem xpord2ind 8172
Description: Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
Hypotheses
Ref Expression
xpord2ind.1 𝑅 Fr 𝐴
xpord2ind.2 𝑅 Po 𝐴
xpord2ind.3 𝑅 Se 𝐴
xpord2ind.4 𝑆 Fr 𝐵
xpord2ind.5 𝑆 Po 𝐵
xpord2ind.6 𝑆 Se 𝐵
xpord2ind.7 (𝑎 = 𝑐 → (𝜑𝜓))
xpord2ind.8 (𝑏 = 𝑑 → (𝜓𝜒))
xpord2ind.9 (𝑎 = 𝑐 → (𝜃𝜒))
xpord2ind.11 (𝑎 = 𝑋 → (𝜑𝜏))
xpord2ind.12 (𝑏 = 𝑌 → (𝜏𝜂))
xpord2ind.i ((𝑎𝐴𝑏𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))
Assertion
Ref Expression
xpord2ind ((𝑋𝐴𝑌𝐵) → 𝜂)
Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑   𝜓,𝑎   𝜏,𝑎   𝐵,𝑎,𝑏,𝑐,𝑑   𝜒,𝑏   𝜂,𝑏   𝜑,𝑐   𝜃,𝑐   𝜓,𝑑   𝑅,𝑎,𝑏,𝑐,𝑑   𝑆,𝑎,𝑏,𝑐,𝑑   𝑋,𝑎,𝑏   𝑌,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑑)   𝜓(𝑏,𝑐)   𝜒(𝑎,𝑐,𝑑)   𝜃(𝑎,𝑏,𝑑)   𝜏(𝑏,𝑐,𝑑)   𝜂(𝑎,𝑐,𝑑)   𝑋(𝑐,𝑑)   𝑌(𝑎,𝑐,𝑑)

Proof of Theorem xpord2ind
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑆(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st𝑥)𝑅(1st𝑦) ∨ (1st𝑥) = (1st𝑦)) ∧ ((2nd𝑥)𝑆(2nd𝑦) ∨ (2nd𝑥) = (2nd𝑦)) ∧ 𝑥𝑦))}
2 xpord2ind.1 . 2 𝑅 Fr 𝐴
3 xpord2ind.2 . 2 𝑅 Po 𝐴
4 xpord2ind.3 . 2 𝑅 Se 𝐴
5 xpord2ind.4 . 2 𝑆 Fr 𝐵
6 xpord2ind.5 . 2 𝑆 Po 𝐵
7 xpord2ind.6 . 2 𝑆 Se 𝐵
8 xpord2ind.7 . 2 (𝑎 = 𝑐 → (𝜑𝜓))
9 xpord2ind.8 . 2 (𝑏 = 𝑑 → (𝜓𝜒))
10 xpord2ind.9 . 2 (𝑎 = 𝑐 → (𝜃𝜒))
11 xpord2ind.11 . 2 (𝑎 = 𝑋 → (𝜑𝜏))
12 xpord2ind.12 . 2 (𝑏 = 𝑌 → (𝜏𝜂))
13 xpord2ind.i . 2 ((𝑎𝐴𝑏𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13xpord2indlem 8171 1 ((𝑋𝐴𝑌𝐵) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059   class class class wbr 5148  {copab 5210   Po wpo 5595   Fr wfr 5638   Se wse 5639   × cxp 5687  Predcpred 6322  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-fr 5641  df-se 5642  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  on2ind  8706  no2indslem  28002
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