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Mirrors > Home > ILE Home > Th. List > ecoptocl | 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 6481 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅) | |
2 | eqid 2139 | . . . . 5 ⊢ (𝐵 × 𝐶) = (𝐵 × 𝐶) | |
3 | eceq1 6464 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → [〈𝑥, 𝑦〉]𝑅 = [𝑧]𝑅) | |
4 | 3 | eqeq2d 2151 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝑧 → (𝐴 = [〈𝑥, 𝑦〉]𝑅 ↔ 𝐴 = [𝑧]𝑅)) |
5 | 4 | imbi1d 230 | . . . . 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 2142 | . . . . . 6 ⊢ (𝐴 = [〈𝑥, 𝑦〉]𝑅 → (𝜑 ↔ 𝜓)) |
9 | 6, 8 | syl5ibcom 154 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → (𝐴 = [〈𝑥, 𝑦〉]𝑅 → 𝜓)) |
10 | 2, 5, 9 | optocl 4615 | . . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅 → 𝜓)) |
11 | 10 | rexlimiv 2543 | . . 3 ⊢ (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅 → 𝜓) |
12 | 1, 11 | syl 14 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓) |
13 | ecoptocl.1 | . 2 ⊢ 𝑆 = ((𝐵 × 𝐶) / 𝑅) | |
14 | 12, 13 | eleq2s 2234 | 1 ⊢ (𝐴 ∈ 𝑆 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 〈cop 3530 × cxp 4537 [cec 6427 / cqs 6428 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 df-qs 6435 |
This theorem is referenced by: 2ecoptocl 6517 3ecoptocl 6518 mulidnq 7204 recexnq 7205 ltsonq 7213 distrnq0 7274 addassnq0 7277 ltposr 7578 0idsr 7582 1idsr 7583 00sr 7584 recexgt0sr 7588 archsr 7597 srpospr 7598 map2psrprg 7620 |
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