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Theorem ecoptocl 8847
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 8810 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2 eqid 2737 . . . . 5 (𝐵 × 𝐶) = (𝐵 × 𝐶)
3 eceq1 8784 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
43eqeq2d 2748 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝐴 = [𝑧]𝑅))
54imbi1d 341 . . . . 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 2745 . . . . . 6 (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑𝜓))
96, 8syl5ibcom 245 . . . . 5 ((𝑥𝐵𝑦𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝜓))
102, 5, 9optocl 5780 . . . 4 (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅𝜓))
1110rexlimiv 3148 . . 3 (∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅𝜓)
121, 11syl 17 . 2 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → 𝜓)
13 ecoptocl.1 . 2 𝑆 = ((𝐵 × 𝐶) / 𝑅)
1412, 13eleq2s 2859 1 (𝐴𝑆𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cop 4632   × cxp 5683  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  2ecoptocl  8848  3ecoptocl  8849  0idsr  11137  1idsr  11138  00sr  11139  recexsrlem  11143  map2psrpr  11150
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