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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 5032 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥)) | |
2 | breq 5032 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦)) | |
3 | 1, 2 | anbi12d 633 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
4 | 3 | exbidv 1922 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
5 | 4 | opabbidv 5096 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}) |
6 | df-coss 35819 | . 2 ⊢ ≀ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} | |
7 | df-coss 35819 | . 2 ⊢ ≀ 𝐵 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2858 | 1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 class class class wbr 5030 {copab 5092 ≀ ccoss 35613 |
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-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-br 5031 df-opab 5093 df-coss 35819 |
This theorem is referenced by: cosseqi 35832 cosseqd 35833 elfunsALTV 36085 |
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