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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosseq | Structured version Visualization version GIF version |
Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.) |
Ref | Expression |
---|---|
cosseq | ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 5150 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥)) | |
2 | breq 5150 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦)) | |
3 | 1, 2 | anbi12d 632 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
4 | 3 | exbidv 1919 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
5 | 4 | opabbidv 5214 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}) |
6 | df-coss 38393 | . 2 ⊢ ≀ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} | |
7 | df-coss 38393 | . 2 ⊢ ≀ 𝐵 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 class class class wbr 5148 {copab 5210 ≀ ccoss 38162 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-br 5149 df-opab 5211 df-coss 38393 |
This theorem is referenced by: cosseqi 38409 cosseqd 38410 elfunsALTV 38674 |
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