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Mirrors > Home > MPE Home > Th. List > llyeq | Structured version Visualization version GIF version |
Description: Equality theorem for the Locally 𝐴 predicate. (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
llyeq | ⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2901 | . . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑢) ∈ 𝐵)) | |
2 | 1 | anbi2d 630 | . . . . 5 ⊢ (𝐴 = 𝐵 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵))) |
3 | 2 | rexbidv 3297 | . . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵))) |
4 | 3 | 2ralbidv 3199 | . . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵))) |
5 | 4 | rabbidv 3480 | . 2 ⊢ (𝐴 = 𝐵 → {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)}) |
6 | df-lly 22068 | . 2 ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} | |
7 | df-lly 22068 | . 2 ⊢ Locally 𝐵 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐵)} | |
8 | 5, 6, 7 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → Locally 𝐴 = Locally 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 ∩ cin 3934 𝒫 cpw 4538 (class class class)co 7150 ↾t crest 16688 Topctop 21495 Locally clly 22066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 df-rab 3147 df-lly 22068 |
This theorem is referenced by: ismntoplly 31261 |
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