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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 7687 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅) | |
2 | eqid 2610 | . . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶) | |
3 | eceq1 7669 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → [〈𝑥, 𝑦〉]𝑅 = [𝑧]𝑅) | |
4 | 3 | eqeq2d 2620 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → (𝐴 = [〈𝑥, 𝑦〉]𝑅 ↔ 𝐴 = [𝑧]𝑅)) |
5 | 4 | imbi1d 330 | . . . . 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 2618 | . . . . . 6 ⊢ (𝐴 = [〈𝑥, 𝑦〉]𝑅 → (𝜑 ↔ 𝜓)) |
9 | 6, 8 | syl5ibcom 234 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓)) |
10 | 2, 5, 9 | optocl 5118 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓)) |
11 | 10 | rexlimiv 3009 | . . 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 2706 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 〈cop 4131 × cxp 5036 [cec 7627 / cqs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 df-qs 7635 |
This theorem is referenced by: 2ecoptocl 7725 3ecoptocl 7726 0idsr 9797 1idsr 9798 00sr 9799 recexsrlem 9803 map2psrpr 9810 |
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