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Theorem xpord2ind 8080
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 2736 . 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 8079 1 ((𝑋𝐴𝑌𝐵) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064   class class class wbr 5105  {copab 5167   Po wpo 5543   Fr wfr 5585   Se wse 5586   × cxp 5631  Predcpred 6252  cfv 6496  1st c1st 7919  2nd c2nd 7920
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-fr 5588  df-se 5589  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922
This theorem is referenced by:  on2ind  8615  no2indslem  27266
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