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Mirrors > Home > MPE Home > Th. List > uniqs2 | Structured version Visualization version GIF version |
Description: The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.) |
Ref | Expression |
---|---|
qsss.1 | ⊢ (𝜑 → 𝑅 Er 𝐴) |
qsss.2 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
Ref | Expression |
---|---|
uniqs2 | ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qsss.2 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
2 | uniqs 8351 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
4 | qsss.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
5 | erdm 8293 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
7 | 6 | imaeq2d 5923 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) |
8 | 3, 7 | eqtr4d 2859 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) |
9 | imadmrn 5933 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
10 | 8, 9 | syl6eq 2872 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) |
11 | errn 8305 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) |
13 | 10, 12 | eqtrd 2856 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cuni 4831 dom cdm 5549 ran crn 5550 “ cima 5552 Er wer 8280 / cqs 8282 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-er 8283 df-ec 8285 df-qs 8289 |
This theorem is referenced by: qshash 15176 cldsubg 22713 pi1buni 23638 qustrivr 30925 |
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