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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qsresid | Structured version Visualization version GIF version |
Description: Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.) |
Ref | Expression |
---|---|
qsresid | ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecres2 37610 | . . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅) | |
2 | 1 | eqeq2d 2742 | . . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅)) |
3 | 2 | rexbiia 3091 | . . 3 ⊢ (∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅) |
4 | 3 | abbii 2801 | . 2 ⊢ {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅} |
5 | df-qs 8715 | . 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} | |
6 | df-qs 8715 | . 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅} | |
7 | 4, 5, 6 | 3eqtr4i 2769 | 1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 {cab 2708 ∃wrex 3069 ↾ cres 5678 [cec 8707 / cqs 8708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ec 8711 df-qs 8715 |
This theorem is referenced by: n0elim 37983 |
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