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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 8771 | . . . . 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 8713 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
7 | 6 | imaeq2d 6060 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) |
8 | 3, 7 | eqtr4d 2776 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) |
9 | imadmrn 6070 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
10 | 8, 9 | eqtrdi 2789 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) |
11 | errn 8725 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) |
13 | 10, 12 | eqtrd 2773 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cuni 4909 dom cdm 5677 ran crn 5678 “ cima 5680 Er wer 8700 / cqs 8702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-er 8703 df-ec 8705 df-qs 8709 |
This theorem is referenced by: qshash 15773 cldsubg 23615 pi1buni 24556 qustrivr 32508 |
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