Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 5042 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑥 ↔ 𝑢𝐵𝑥)) | |
2 | breq 5042 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑢𝐴𝑦 ↔ 𝑢𝐵𝑦)) | |
3 | 1, 2 | anbi12d 634 | . . . 4 ⊢ (𝐴 = 𝐵 → ((𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ (𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
4 | 3 | exbidv 1928 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦) ↔ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦))) |
5 | 4 | opabbidv 5106 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)}) |
6 | df-coss 36192 | . 2 ⊢ ≀ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐴𝑥 ∧ 𝑢𝐴𝑦)} | |
7 | df-coss 36192 | . 2 ⊢ ≀ 𝐵 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝐵𝑥 ∧ 𝑢𝐵𝑦)} | |
8 | 5, 6, 7 | 3eqtr4g 2799 | 1 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∃wex 1786 class class class wbr 5040 {copab 5102 ≀ ccoss 35988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-br 5041 df-opab 5103 df-coss 36192 |
This theorem is referenced by: cosseqi 36205 cosseqd 36206 elfunsALTV 36458 |
Copyright terms: Public domain | W3C validator |