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Mirrors > Home > MPE Home > Th. List > uniqs | Structured version Visualization version GIF version |
Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.) |
Ref | Expression |
---|---|
uniqs | ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecexg 8653 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V) | |
2 | 1 | ralrimivw 3148 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V) |
3 | dfiun2g 4991 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) |
5 | 4 | eqcomd 2743 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅) |
6 | df-qs 8655 | . . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
7 | 6 | unieqi 4879 | . 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
8 | df-ec 8651 | . . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥}) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥})) |
10 | 9 | iuneq2i 4976 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) |
11 | imaiun 7193 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) | |
12 | iunid 5021 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
13 | 12 | imaeq2i 6012 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴) |
14 | 10, 11, 13 | 3eqtr2ri 2772 | . 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 |
15 | 5, 7, 14 | 3eqtr4g 2802 | 1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 ∃wrex 3074 Vcvv 3446 {csn 4587 ∪ cuni 4866 ∪ ciun 4955 “ cima 5637 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ec 8651 df-qs 8655 |
This theorem is referenced by: uniqs2 8719 ecqs 8721 |
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