| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj602 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj602 | ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq2 5147 | . . 3 ⊢ (𝑋 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑦𝑅𝑌)) | |
| 2 | 1 | rabbidv 3444 | . 2 ⊢ (𝑋 = 𝑌 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌}) |
| 3 | df-bnj14 34703 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
| 4 | df-bnj14 34703 | . 2 ⊢ pred(𝑌, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌} | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {crab 3436 class class class wbr 5143 predc-bnj14 34702 |
| 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 |
| 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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-bnj14 34703 |
| This theorem is referenced by: bnj601 34934 bnj852 34935 bnj18eq1 34941 bnj938 34951 bnj1125 35006 bnj1148 35010 bnj1318 35039 bnj1442 35063 bnj1450 35064 bnj1452 35066 bnj1463 35069 bnj1529 35084 |
| Copyright terms: Public domain | W3C validator |