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| Mirrors > Home > MPE Home > Th. List > Mathboxes > trressn | Structured version Visualization version GIF version | ||
| Description: Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38443) is transitive, see also trrelressn 38584. (Contributed by Peter Mazsa, 16-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| trressn | ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | an3 659 | . . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧)) | |
| 2 | eqbrb 38234 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑦)) | |
| 3 | eqbrb 38234 | . . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦𝑅𝑧) ↔ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧)) | |
| 4 | 2, 3 | anbi12i 628 | . . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) ↔ ((𝑥 = 𝐴 ∧ 𝐴𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝐴𝑅𝑧))) | 
| 5 | eqbrb 38234 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑧) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑧)) | |
| 6 | 1, 4, 5 | 3imtr4i 292 | . . . 4 ⊢ (((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) → (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧)) | 
| 7 | brressn 38442 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦))) | |
| 8 | 7 | el2v 3487 | . . . . 5 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑦 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑦)) | 
| 9 | brressn 38442 | . . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧))) | |
| 10 | 9 | el2v 3487 | . . . . 5 ⊢ (𝑦(𝑅 ↾ {𝐴})𝑧 ↔ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧)) | 
| 11 | 8, 10 | anbi12i 628 | . . . 4 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) ↔ ((𝑥 = 𝐴 ∧ 𝑥𝑅𝑦) ∧ (𝑦 = 𝐴 ∧ 𝑦𝑅𝑧))) | 
| 12 | brressn 38442 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧))) | |
| 13 | 12 | el2v 3487 | . . . 4 ⊢ (𝑥(𝑅 ↾ {𝐴})𝑧 ↔ (𝑥 = 𝐴 ∧ 𝑥𝑅𝑧)) | 
| 14 | 6, 11, 13 | 3imtr4i 292 | . . 3 ⊢ ((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) | 
| 15 | 14 | gen2 1796 | . 2 ⊢ ∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) | 
| 16 | 15 | ax-gen 1795 | 1 ⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 Vcvv 3480 {csn 4626 class class class wbr 5143 ↾ cres 5687 | 
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 | 
| This theorem is referenced by: trrelressn 38584 | 
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