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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 8783 | . . . . 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 5239 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))) |
5 | 4 | oveq1d 7417 | . 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 2724 | . . 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 17493 | . 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
12 | eqidd 2725 | . . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) | |
13 | eqid 2724 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
14 | inex1g 5310 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) | |
15 | 9, 14 | syl 17 | . . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) |
16 | 12, 7, 13, 15, 10 | qusval 17493 | . 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅)) |
17 | 5, 11, 16 | 3eqtr4d 2774 | 1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∩ cin 3940 ⊆ wss 3941 ↦ cmpt 5222 × cxp 5665 “ cima 5670 ‘cfv 6534 (class class class)co 7402 [cec 8698 Basecbs 17149 “s cimas 17455 /s cqus 17456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-ec 8702 df-qus 17460 |
This theorem is referenced by: pi1addf 24918 pi1addval 24919 pi1grplem 24920 |
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