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Theorem xpord2ind 8085
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 2737 . 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 8084 1 ((𝑋𝐴𝑌𝐵) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065   class class class wbr 5110  {copab 5172   Po wpo 5548   Fr wfr 5590   Se wse 5591   × cxp 5636  Predcpred 6257  cfv 6501  1st c1st 7924  2nd c2nd 7925
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-po 5550  df-fr 5593  df-se 5594  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fun 6503  df-fv 6509  df-1st 7926  df-2nd 7927
This theorem is referenced by:  on2ind  8620  no2indslem  27288
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