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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 8360 | . . . . 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 8302 | . . . . . 6 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝑅 = 𝐴) |
7 | 6 | imaeq2d 5932 | . . . 4 ⊢ (𝜑 → (𝑅 “ dom 𝑅) = (𝑅 “ 𝐴)) |
8 | 3, 7 | eqtr4d 2862 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = (𝑅 “ dom 𝑅)) |
9 | imadmrn 5942 | . . 3 ⊢ (𝑅 “ dom 𝑅) = ran 𝑅 | |
10 | 8, 9 | syl6eq 2875 | . 2 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = ran 𝑅) |
11 | errn 8314 | . . 3 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴) | |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → ran 𝑅 = 𝐴) |
13 | 10, 12 | eqtrd 2859 | 1 ⊢ (𝜑 → ∪ (𝐴 / 𝑅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∪ cuni 4841 dom cdm 5558 ran crn 5559 “ cima 5561 Er wer 8289 / cqs 8291 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-er 8292 df-ec 8294 df-qs 8298 |
This theorem is referenced by: qshash 15185 cldsubg 22722 pi1buni 23647 qustrivr 30934 |
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