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| Mirrors > Home > MPE Home > Th. List > xpord2ind | Structured version Visualization version GIF version | ||
| 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.) | 
| 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 → (𝜑 ↔ 𝜓)) | 
| xpord2ind.8 | ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒)) | 
| xpord2ind.9 | ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒)) | 
| xpord2ind.11 | ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏)) | 
| xpord2ind.12 | ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂)) | 
| xpord2ind.i | ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑)) | 
| Ref | Expression | 
|---|---|
| xpord2ind | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))} | |
| 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 → (𝜑 ↔ 𝜓)) | |
| 9 | xpord2ind.8 | . 2 ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒)) | |
| 10 | xpord2ind.9 | . 2 ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒)) | |
| 11 | xpord2ind.11 | . 2 ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏)) | |
| 12 | xpord2ind.12 | . 2 ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂)) | |
| 13 | xpord2ind.i | . 2 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑)) | |
| 14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | xpord2indlem 8172 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 class class class wbr 5143 {copab 5205 Po wpo 5590 Fr wfr 5634 Se wse 5635 × cxp 5683 Predcpred 6320 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-fr 5637 df-se 5638 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: on2ind 8707 no2indslem 27987 | 
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