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Mirrors > Home > MPE Home > Th. List > Mathboxes > uniqsALTV | Structured version Visualization version GIF version |
Description: The union of a quotient set: a weaker version of uniqs 7807. (Contributed by Peter Mazsa, 20-Jun-2019.) |
Ref | Expression |
---|---|
uniqsALTV | ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecex2 34100 | . . . . 5 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝑥 ∈ 𝐴 → [𝑥]𝑅 ∈ V)) | |
2 | 1 | ralrimiv 2965 | . . . 4 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V) |
3 | dfiun2g 4552 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 [𝑥]𝑅 ∈ V → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}) |
5 | 4 | eqcomd 2628 | . 2 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅) |
6 | df-qs 7748 | . . 3 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
7 | 6 | unieqi 4445 | . 2 ⊢ ∪ (𝐴 / 𝑅) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
8 | df-ec 7744 | . . . . 5 ⊢ [𝑥]𝑅 = (𝑅 “ {𝑥}) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥})) |
10 | 9 | iuneq2i 4539 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) |
11 | imaiun 6503 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = ∪ 𝑥 ∈ 𝐴 (𝑅 “ {𝑥}) | |
12 | iunid 4575 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
13 | 12 | imaeq2i 5464 | . . 3 ⊢ (𝑅 “ ∪ 𝑥 ∈ 𝐴 {𝑥}) = (𝑅 “ 𝐴) |
14 | 10, 11, 13 | 3eqtr2ri 2651 | . 2 ⊢ (𝑅 “ 𝐴) = ∪ 𝑥 ∈ 𝐴 [𝑥]𝑅 |
15 | 5, 7, 14 | 3eqtr4g 2681 | 1 ⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 Vcvv 3200 {csn 4177 ∪ cuni 4436 ∪ ciun 4520 ↾ cres 5116 “ cima 5117 [cec 7740 / cqs 7741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 df-qs 7748 |
This theorem is referenced by: rnresequniqs 34102 |
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