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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 7807 | . . . . 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 7752 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
7 | 6 | imaeq2d 5466 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) |
8 | 3, 7 | eqtr4d 2659 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) |
9 | imadmrn 5476 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
10 | 8, 9 | syl6eq 2672 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) |
11 | errn 7764 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) |
13 | 10, 12 | eqtrd 2656 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 dom cdm 5114 ran crn 5115 “ cima 5117 Er wer 7739 / 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-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-er 7742 df-ec 7744 df-qs 7748 |
This theorem is referenced by: qshash 14559 cldsubg 21914 pi1buni 22840 |
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