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Mirrors > Home > MPE Home > Th. List > Mathboxes > crefeq | Structured version Visualization version GIF version |
Description: Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
crefeq | ⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq2 3808 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝑗 ∩ 𝐵)) | |
2 | 1 | rexeqdv 3145 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)) |
3 | 2 | imbi2d 330 | . . . 4 ⊢ (𝐴 = 𝐵 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))) |
4 | 3 | ralbidv 2986 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))) |
5 | 4 | rabbidv 3189 | . 2 ⊢ (𝐴 = 𝐵 → {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)}) |
6 | df-cref 29910 | . 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} | |
7 | df-cref 29910 | . 2 ⊢ CovHasRef𝐵 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∀wral 2912 ∃wrex 2913 {crab 2916 ∩ cin 3573 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 Topctop 20698 Refcref 21305 CovHasRefccref 29909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-in 3581 df-cref 29910 |
This theorem is referenced by: ispcmp 29924 |
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