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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 8374 | . . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
3 | 1, 2 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) |
4 | 3 | mpteq2dva 5163 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))) |
5 | 4 | oveq1d 7173 | . 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 2823 | . . 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 16817 | . 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
12 | eqidd 2824 | . . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) | |
13 | eqid 2823 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
14 | inex1g 5225 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) | |
15 | 9, 14 | syl 17 | . . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) |
16 | 12, 7, 13, 15, 10 | qusval 16817 | . 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅)) |
17 | 5, 11, 16 | 3eqtr4d 2868 | 1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ↦ cmpt 5148 × cxp 5555 “ cima 5560 ‘cfv 6357 (class class class)co 7158 [cec 8289 Basecbs 16485 “s cimas 16779 /s cqus 16780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-ec 8293 df-qus 16784 |
This theorem is referenced by: pi1addf 23653 pi1addval 23654 pi1grplem 23655 |
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