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Mirrors > Home > MPE Home > Th. List > quslem | Structured version Visualization version GIF version |
Description: The function in qusval 16025 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 7633 | . . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V) |
4 | 3 | ralrimivw 2950 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V) |
5 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
6 | 5 | fnmpt 5933 | . . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
8 | dffn4 6034 | . . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹) | |
9 | 7, 8 | sylib 207 | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹) |
10 | 5 | rnmpt 5292 | . . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } |
11 | df-qs 7635 | . . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } | |
12 | 10, 11 | eqtr4i 2635 | . . 3 ⊢ ran 𝐹 = (𝑉 / ∼ ) |
13 | foeq3 6026 | . . 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 207 | 1 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ↦ cmpt 4643 ran crn 5039 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 [cec 7627 / cqs 7628 Basecbs 15695 /s cqus 15988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-fo 5810 df-ec 7631 df-qs 7635 |
This theorem is referenced by: qusbas 16028 quss 16029 qusaddvallem 16034 qusaddflem 16035 qusaddval 16036 qusaddf 16037 qusmulval 16038 qusmulf 16039 qusgrp2 17356 qusring2 18443 znzrhfo 19715 qustps 21335 qustgpopn 21733 qustgplem 21734 qustgphaus 21736 |
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