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Theorem ecoptocl 8418
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 8381 . . 3 (𝐴 ∈ ((𝐵 × 𝐶) / 𝑅) → ∃𝑧 ∈ (𝐵 × 𝐶)𝐴 = [𝑧]𝑅)
2 eqid 2738 . . . . 5 (𝐵 × 𝐶) = (𝐵 × 𝐶)
3 eceq1 8358 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝑧 → [⟨𝑥, 𝑦⟩]𝑅 = [𝑧]𝑅)
43eqeq2d 2749 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝑧 → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝐴 = [𝑧]𝑅))
54imbi1d 345 . . . . 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 2746 . . . . . 6 (𝐴 = [⟨𝑥, 𝑦⟩]𝑅 → (𝜑𝜓))
96, 8syl5ibcom 248 . . . . 5 ((𝑥𝐵𝑦𝐶) → (𝐴 = [⟨𝑥, 𝑦⟩]𝑅𝜓))
102, 5, 9optocl 5616 . . . 4 (𝑧 ∈ (𝐵 × 𝐶) → (𝐴 = [𝑧]𝑅𝜓))
1110rexlimiv 3190 . . 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 2851 1 (𝐴𝑆𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  cop 4522   × cxp 5523  [cec 8318   / cqs 8319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ec 8322  df-qs 8326
This theorem is referenced by:  2ecoptocl  8419  3ecoptocl  8420  0idsr  10597  1idsr  10598  00sr  10599  recexsrlem  10603  map2psrpr  10610
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