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Mirrors > Home > MPE Home > Th. List > quslem | Structured version Visualization version GIF version |
Description: The function in qusval 17589 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qusval.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) |
qusval.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
qusval.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
qusval.e | ⊢ (𝜑 → ∼ ∈ 𝑊) |
qusval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
Ref | Expression |
---|---|
quslem | ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusval.e | . . . . . 6 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
2 | ecexg 8748 | . . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V) |
4 | 3 | ralrimivw 3148 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V) |
5 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
6 | 5 | fnmpt 6709 | . . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
8 | dffn4 6827 | . . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹) | |
9 | 7, 8 | sylib 218 | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹) |
10 | 5 | rnmpt 5971 | . . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } |
11 | df-qs 8750 | . . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } | |
12 | 10, 11 | eqtr4i 2766 | . . 3 ⊢ ran 𝐹 = (𝑉 / ∼ ) |
13 | foeq3 6819 | . . 3 ⊢ (ran 𝐹 = (𝑉 / ∼ ) → (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ ))) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ )) |
15 | 9, 14 | sylib 218 | 1 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 [cec 8742 / cqs 8743 Basecbs 17245 /s cqus 17552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-fo 6569 df-ec 8746 df-qs 8750 |
This theorem is referenced by: qusbas 17592 quss 17593 qusaddvallem 17598 qusaddflem 17599 qusaddval 17600 qusaddf 17601 qusmulval 17602 qusmulf 17603 qusgrp2 19089 qusrng 20198 qusring2 20348 znzrhfo 21584 qustps 23746 qustgpopn 24144 qustgplem 24145 qustgphaus 24147 qusker 33357 qusvsval 33360 quslmod 33366 quslmhm 33367 qusdimsum 33656 |
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