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Mirrors > Home > ILE Home > Th. List > uniqs | 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 6538 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → [𝑥]𝑅 ∈ V) | |
2 | 1 | ralrimivw 2551 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V) |
3 | dfiun2g 3918 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) |
5 | 4 | eqcomd 2183 | . 2 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅) |
6 | df-qs 6540 | . . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
7 | 6 | unieqi 3819 | . 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
8 | df-ec 6536 | . . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥}) | |
9 | 8 | a1i 9 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥})) |
10 | 9 | iuneq2i 3904 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) |
11 | imaiun 5760 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) | |
12 | iunid 3942 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
13 | 12 | imaeq2i 4968 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴) |
14 | 10, 11, 13 | 3eqtr2ri 2205 | . 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 |
15 | 5, 7, 14 | 3eqtr4g 2235 | 1 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 ∃wrex 2456 Vcvv 2737 {csn 3592 ∪ cuni 3809 ∪ ciun 3886 “ cima 4629 [cec 6532 / cqs 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-ec 6536 df-qs 6540 |
This theorem is referenced by: uniqs2 6594 ecqs 6596 |
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