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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 8817 | . . . . 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 8755 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
| 6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) | 
| 7 | 6 | imaeq2d 6078 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) | 
| 8 | 3, 7 | eqtr4d 2780 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) | 
| 9 | imadmrn 6088 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
| 10 | 8, 9 | eqtrdi 2793 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) | 
| 11 | errn 8767 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
| 12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) | 
| 13 | 10, 12 | eqtrd 2777 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 dom cdm 5685 ran crn 5686 “ cima 5688 Er wer 8742 / cqs 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-er 8745 df-ec 8747 df-qs 8751 | 
| This theorem is referenced by: qshash 15863 cldsubg 24119 pi1buni 25073 qustrivr 33393 | 
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