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Mirrors > Home > MPE Home > Th. List > qusin | Structured version Visualization version GIF version |
Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qusin.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) |
qusin.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
qusin.e | ⊢ (𝜑 → ∼ ∈ 𝑊) |
qusin.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
qusin.s | ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉) |
Ref | Expression |
---|---|
qusin | ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusin.s | . . . . 5 ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉) | |
2 | ecinxp 8810 | . . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
3 | 1, 2 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) |
4 | 3 | mpteq2dva 5248 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))) |
5 | 4 | oveq1d 7435 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅)) |
6 | qusin.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
7 | qusin.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | eqid 2728 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
9 | qusin.e | . . 3 ⊢ (𝜑 → ∼ ∈ 𝑊) | |
10 | qusin.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
11 | 6, 7, 8, 9, 10 | qusval 17523 | . 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
12 | eqidd 2729 | . . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) | |
13 | eqid 2728 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
14 | inex1g 5319 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) | |
15 | 9, 14 | syl 17 | . . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) |
16 | 12, 7, 13, 15, 10 | qusval 17523 | . 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅)) |
17 | 5, 11, 16 | 3eqtr4d 2778 | 1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ↦ cmpt 5231 × cxp 5676 “ cima 5681 ‘cfv 6548 (class class class)co 7420 [cec 8722 Basecbs 17179 “s cimas 17485 /s cqus 17486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-ec 8726 df-qus 17490 |
This theorem is referenced by: pi1addf 24973 pi1addval 24974 pi1grplem 24975 |
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