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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 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 = 𝐵) |
3 | 2 | raleqdv 3326 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦))) |
4 | 1, 3 | rabeqbidva 3449 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)}) |
5 | df-scott 42928 | . 2 ⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
6 | df-scott 42928 | . 2 ⊢ Scott 𝐵 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 (rank‘𝑥) ⊆ (rank‘𝑦)} | |
7 | 4, 5, 6 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → Scott 𝐴 = Scott 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 ⊆ wss 3947 ‘cfv 6540 rankcrnk 9754 Scott cscott 42927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-scott 42928 |
This theorem is referenced by: scotteq 42930 dfcoll2 42944 colleq12d 42945 |
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