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Theorem ecoptocl 6619
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 6584 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2 eqid 2177 . . . . 5 (𝐵 × 𝐶) = (𝐵 × 𝐶)
3 eceq1 6567 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
43eqeq2d 2189 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝐴 = [𝑧]𝑅))
54imbi1d 231 . . . . 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 2180 . . . . . 6 (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑𝜓))
96, 8syl5ibcom 155 . . . . 5 ((𝑥𝐵𝑦𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝜓))
102, 5, 9optocl 4701 . . . 4 (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅𝜓))
1110rexlimiv 2588 . . 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 2272 1 (𝐴𝑆𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456  cop 3595   × cxp 4623  [cec 6530   / cqs 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-ec 6534  df-qs 6538
This theorem is referenced by:  2ecoptocl  6620  3ecoptocl  6621  mulidnq  7385  recexnq  7386  ltsonq  7394  distrnq0  7455  addassnq0  7458  ltposr  7759  0idsr  7763  1idsr  7764  00sr  7765  recexgt0sr  7769  archsr  7778  srpospr  7779  map2psrprg  7801
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