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| Mirrors > Home > MPE Home > Th. List > quslem | Structured version Visualization version GIF version | ||
| Description: The function in qusval 17587 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 8749 | . . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V) |
| 4 | 3 | ralrimivw 3150 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V) |
| 5 | qusval.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
| 6 | 5 | fnmpt 6708 | . . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉) |
| 7 | 4, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
| 8 | dffn4 6826 | . . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹) | |
| 9 | 7, 8 | sylib 218 | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹) |
| 10 | 5 | rnmpt 5968 | . . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } |
| 11 | df-qs 8751 | . . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ } | |
| 12 | 10, 11 | eqtr4i 2768 | . . 3 ⊢ ran 𝐹 = (𝑉 / ∼ ) |
| 13 | foeq3 6818 | . . 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 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ↦ cmpt 5225 ran crn 5686 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 [cec 8743 / cqs 8744 Basecbs 17247 /s cqus 17550 |
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-fo 6567 df-ec 8747 df-qs 8751 |
| This theorem is referenced by: qusbas 17590 quss 17591 qusaddvallem 17596 qusaddflem 17597 qusaddval 17598 qusaddf 17599 qusmulval 17600 qusmulf 17601 qusgrp2 19076 qusrng 20177 qusring2 20331 znzrhfo 21566 qustps 23730 qustgpopn 24128 qustgplem 24129 qustgphaus 24131 qusker 33377 qusvsval 33380 quslmod 33386 quslmhm 33387 qusdimsum 33679 |
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