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Mirrors > Home > MPE Home > Th. List > ecoptocl | Structured version Visualization version GIF version |
Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
ecoptocl.1 | ⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅) |
ecoptocl.2 | ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) |
ecoptocl.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) |
Ref | Expression |
---|---|
ecoptocl | ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsi 8789 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅) | |
2 | eqid 2725 | . . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶) | |
3 | eceq1 8763 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → [〈𝑥, 𝑦〉]𝑅 = [𝑧]𝑅) | |
4 | 3 | eqeq2d 2736 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → (𝐴 = [〈𝑥, 𝑦〉]𝑅 ↔ 𝐴 = [𝑧]𝑅)) |
5 | 4 | imbi1d 340 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → ((𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓) ↔ (𝐴 = [𝑧]𝑅 → 𝜓))) |
6 | ecoptocl.3 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) | |
7 | ecoptocl.2 | . . . . . . 7 ⊢ ([〈𝑥, 𝑦〉]𝑅 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 7 | eqcoms 2733 | . . . . . 6 ⊢ (𝐴 = [〈𝑥, 𝑦〉]𝑅 → (𝜑 ↔ 𝜓)) |
9 | 6, 8 | syl5ibcom 244 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓)) |
10 | 2, 5, 9 | optocl 5772 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓)) |
11 | 10 | rexlimiv 3137 | . . 3 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅 → 𝜓) |
12 | 1, 11 | syl 17 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓) |
13 | ecoptocl.1 | . 2 ⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅) | |
14 | 12, 13 | eleq2s 2843 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 〈cop 4636 × cxp 5676 [cec 8723 / cqs 8724 |
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-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 df-qs 8731 |
This theorem is referenced by: 2ecoptocl 8827 3ecoptocl 8828 0idsr 11122 1idsr 11123 00sr 11124 recexsrlem 11128 map2psrpr 11135 |
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