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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 5079 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥)) | |
2 | breq 5079 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦)) | |
3 | 1, 2 | anbi12d 630 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
4 | 3 | exbidv 1920 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
5 | 4 | opabbidv 5143 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}) |
6 | df-coss 36563 | . 2 ⊢ ≀ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} | |
7 | df-coss 36563 | . 2 ⊢ ≀ 𝐵 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1777 class class class wbr 5077 {copab 5139 ≀ ccoss 36361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1778 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-br 5078 df-opab 5140 df-coss 36563 |
This theorem is referenced by: cosseqi 36576 cosseqd 36577 elfunsALTV 36829 |
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