![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8748 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V) | |
2 | 1 | ralrimivw 3148 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V) |
3 | dfiun2g 5035 | . . . 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 2741 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅) |
6 | df-qs 8750 | . . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
7 | 6 | unieqi 4924 | . 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
8 | df-ec 8746 | . . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥}) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥})) |
10 | 9 | iuneq2i 5018 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) |
11 | imaiun 7265 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) | |
12 | iunid 5065 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
13 | 12 | imaeq2i 6078 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴) |
14 | 10, 11, 13 | 3eqtr2ri 2770 | . 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 |
15 | 5, 7, 14 | 3eqtr4g 2800 | 1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 {csn 4631 ∪ cuni 4912 ∪ ciun 4996 “ cima 5692 [cec 8742 / cqs 8743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 |
This theorem is referenced by: uniqs2 8818 ecqs 8820 |
Copyright terms: Public domain | W3C validator |