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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 8350 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅) | |
2 | eqid 2821 | . . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶) | |
3 | eceq1 8327 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → [〈𝑥, 𝑦〉]𝑅 = [𝑧]𝑅) | |
4 | 3 | eqeq2d 2832 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → (𝐴 = [〈𝑥, 𝑦〉]𝑅 ↔ 𝐴 = [𝑧]𝑅)) |
5 | 4 | imbi1d 344 | . . . . 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 2829 | . . . . . 6 ⊢ (𝐴 = [〈𝑥, 𝑦〉]𝑅 → (𝜑 ↔ 𝜓)) |
9 | 6, 8 | syl5ibcom 247 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓)) |
10 | 2, 5, 9 | optocl 5645 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓)) |
11 | 10 | rexlimiv 3280 | . . 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 2931 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 〈cop 4573 × cxp 5553 [cec 8287 / cqs 8288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ec 8291 df-qs 8295 |
This theorem is referenced by: 2ecoptocl 8388 3ecoptocl 8389 0idsr 10519 1idsr 10520 00sr 10521 recexsrlem 10525 map2psrpr 10532 |
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