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Mirrors > Home > ILE Home > Th. List > qsinxp | GIF version |
Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qsinxp | ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecinxp 6588 | . . . . 5 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝑥]𝑅 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))) | |
2 | 1 | eqeq2d 2182 | . . . 4 ⊢ (((𝑅 “ 𝐴) ⊆ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = [𝑥]𝑅 ↔ 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))) |
3 | 2 | rexbidva 2467 | . . 3 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴)))) |
4 | 3 | abbidv 2288 | . 2 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))}) |
5 | df-qs 6519 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
6 | df-qs 6519 | . 2 ⊢ (𝐴 / (𝑅 ∩ (𝐴 × 𝐴))) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥](𝑅 ∩ (𝐴 × 𝐴))} | |
7 | 4, 5, 6 | 3eqtr4g 2228 | 1 ⊢ ((𝑅 “ 𝐴) ⊆ 𝐴 → (𝐴 / 𝑅) = (𝐴 / (𝑅 ∩ (𝐴 × 𝐴)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {cab 2156 ∃wrex 2449 ∩ cin 3120 ⊆ wss 3121 × cxp 4609 “ cima 4614 [cec 6511 / cqs 6512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-ec 6515 df-qs 6519 |
This theorem is referenced by: (None) |
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