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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 8381 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅) | |
2 | eqid 2738 | . . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶) | |
3 | eceq1 8358 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → [〈𝑥, 𝑦〉]𝑅 = [𝑧]𝑅) | |
4 | 3 | eqeq2d 2749 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → (𝐴 = [〈𝑥, 𝑦〉]𝑅 ↔ 𝐴 = [𝑧]𝑅)) |
5 | 4 | imbi1d 345 | . . . . 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 2746 | . . . . . 6 ⊢ (𝐴 = [〈𝑥, 𝑦〉]𝑅 → (𝜑 ↔ 𝜓)) |
9 | 6, 8 | syl5ibcom 248 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓)) |
10 | 2, 5, 9 | optocl 5616 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓)) |
11 | 10 | rexlimiv 3190 | . . 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 2851 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 〈cop 4522 × cxp 5523 [cec 8318 / cqs 8319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ec 8322 df-qs 8326 |
This theorem is referenced by: 2ecoptocl 8419 3ecoptocl 8420 0idsr 10597 1idsr 10598 00sr 10599 recexsrlem 10603 map2psrpr 10610 |
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