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Mirrors > Home > ILE Home > Th. List > uniqs2 | 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 6455 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) |
4 | qsss.1 | . . . . . 6 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
5 | erdm 6407 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
6 | 4, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
7 | 6 | imaeq2d 4851 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) |
8 | 3, 7 | eqtr4d 2153 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) |
9 | imadmrn 4861 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
10 | 8, 9 | syl6eq 2166 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) |
11 | errn 6419 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
12 | 4, 11 | syl 14 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) |
13 | 10, 12 | eqtrd 2150 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∪ cuni 3706 dom cdm 4509 ran crn 4510 “ cima 4512 Er wer 6394 / cqs 6396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-er 6397 df-ec 6399 df-qs 6403 |
This theorem is referenced by: (None) |
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