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| Mirrors > Home > ILE Home > Th. List > qusin | 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 6696 | . . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
| 3 | 1, 2 | sylan 283 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉))) |
| 4 | 3 | mpteq2dva 4133 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))) |
| 5 | 4 | oveq1d 5958 | . 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 2204 | . . 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 13097 | . 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
| 12 | eqidd 2205 | . . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) | |
| 13 | eqid 2204 | . . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) | |
| 14 | inex1g 4179 | . . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) | |
| 15 | 9, 14 | syl 14 | . . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V) |
| 16 | 12, 7, 13, 15, 10 | qusval 13097 | . 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅)) |
| 17 | 5, 11, 16 | 3eqtr4d 2247 | 1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉)))) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 = wceq 1372 ∈ wcel 2175 Vcvv 2771 ∩ cin 3164 ⊆ wss 3165 ↦ cmpt 4104 × cxp 4672 “ cima 4677 ‘cfv 5270 (class class class)co 5943 [cec 6617 Basecbs 12774 “s cimas 13073 /s cqus 13074 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1re 8018 ax-addrcl 8021 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-tp 3640 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-ov 5946 df-oprab 5947 df-mpo 5948 df-ec 6621 df-inn 9036 df-2 9094 df-3 9095 df-ndx 12777 df-slot 12778 df-base 12780 df-plusg 12864 df-mulr 12865 df-iimas 13076 df-qus 13077 |
| This theorem is referenced by: (None) |
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