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Theorem ecoptocl 6282
Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
ecoptocl.1 𝑆 = ((𝐵 × 𝐶) / 𝑅)
ecoptocl.2 ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑𝜓))
ecoptocl.3 ((𝑥𝐵𝑦𝐶) → 𝜑)
Assertion
Ref Expression
ecoptocl (𝐴𝑆𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑅,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem ecoptocl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elqsi 6247 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2 eqid 2083 . . . . 5 (𝐵 × 𝐶) = (𝐵 × 𝐶)
3 eceq1 6230 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
43eqeq2d 2094 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝐴 = [𝑧]𝑅))
54imbi1d 229 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝑧 → ((𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝜓) ↔ (𝐴 = [𝑧]𝑅𝜓)))
6 ecoptocl.3 . . . . . 6 ((𝑥𝐵𝑦𝐶) → 𝜑)
7 ecoptocl.2 . . . . . . 7 ([⟨𝑥, 𝑦⟩]𝑅 = 𝐴 → (𝜑𝜓))
87eqcoms 2086 . . . . . 6 (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑𝜓))
96, 8syl5ibcom 153 . . . . 5 ((𝑥𝐵𝑦𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝜓))
102, 5, 9optocl 4463 . . . 4 (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅𝜓))
1110rexlimiv 2477 . . 3 (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅𝜓)
121, 11syl 14 . 2 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓)
13 ecoptocl.1 . 2 𝑆 = ((𝐵 × 𝐶) / 𝑅)
1412, 13eleq2s 2177 1 (𝐴𝑆𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wrex 2354  cop 3420   × cxp 4390  [cec 6193   / cqs 6194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-ec 6197  df-qs 6201
This theorem is referenced by:  2ecoptocl  6283  3ecoptocl  6284  mulidnq  6718  recexnq  6719  ltsonq  6727  distrnq0  6788  addassnq0  6791  ltposr  7079  0idsr  7083  1idsr  7084  00sr  7085  recexgt0sr  7089  archsr  7097  srpospr  7098
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