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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > scotteqd | Structured version Visualization version GIF version |
Description: Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
scotteqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
scotteqd | ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scotteqd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵) |
3 | 2 | raleqdv 3364 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦))) |
4 | 1, 3 | rabeqbidva 3434 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
5 | df-scott 40944 | . 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
6 | df-scott 40944 | . 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
7 | 4, 5, 6 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 ‘cfv 6324 rankcrnk 9176 Scott cscott 40943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rab 3115 df-scott 40944 |
This theorem is referenced by: scotteq 40946 dfcoll2 40960 colleq12d 40961 |
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