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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 37783 | . . . . 5 ⊢ (𝑣 ∈ 𝐴 → [𝑣](𝑅 ↾ 𝐴) = [𝑣]𝑅) | |
2 | 1 | eqeq2d 2739 | . . . 4 ⊢ (𝑣 ∈ 𝐴 → (𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ 𝑢 = [𝑣]𝑅)) |
3 | 2 | rexbiia 3089 | . . 3 ⊢ (∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴) ↔ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅) |
4 | 3 | abbii 2798 | . 2 ⊢ {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅} |
5 | df-qs 8737 | . 2 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣](𝑅 ↾ 𝐴)} | |
6 | df-qs 8737 | . 2 ⊢ (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣 ∈ 𝐴 𝑢 = [𝑣]𝑅} | |
7 | 4, 5, 6 | 3eqtr4i 2766 | 1 ⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2705 ∃wrex 3067 ↾ cres 5684 [cec 8729 / cqs 8730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ec 8733 df-qs 8737 |
This theorem is referenced by: n0elim 38154 |
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