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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 5059 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥)) | |
2 | breq 5059 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦)) | |
3 | 1, 2 | anbi12d 630 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
4 | 3 | exbidv 1913 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
5 | 4 | opabbidv 5123 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}) |
6 | df-coss 35539 | . 2 ⊢ ≀ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} | |
7 | df-coss 35539 | . 2 ⊢ ≀ 𝐵 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2878 | 1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 class class class wbr 5057 {copab 5119 ≀ ccoss 35334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-br 5058 df-opab 5120 df-coss 35539 |
This theorem is referenced by: cosseqi 35552 cosseqd 35553 elfunsALTV 35805 |
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