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Theorem elmptrab2OLD 21441
 Description: Obsolete version of elmptrab2 21442 as of 26-Mar-2021. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elmptrab2.f 𝐹 = (𝑥 ∈ V ↦ {𝑦𝐵𝜑})
elmptrab2.s1 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
elmptrab2.s2 (𝑥 = 𝑋𝐵 = 𝐶)
elmptrab2OLD.ex 𝐵𝑉
elmptrab2OLD.rc (𝑌𝐶𝑋𝑊)
Assertion
Ref Expression
elmptrab2OLD (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐶,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem elmptrab2OLD
StepHypRef Expression
1 elmptrab2.f . . 3 𝐹 = (𝑥 ∈ V ↦ {𝑦𝐵𝜑})
2 elmptrab2.s1 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
3 elmptrab2.s2 . . 3 (𝑥 = 𝑋𝐵 = 𝐶)
4 elmptrab2OLD.ex . . . 4 𝐵𝑉
54a1i 11 . . 3 (𝑥 ∈ V → 𝐵𝑉)
61, 2, 3, 5elmptrab 21440 . 2 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
7 3simpc 1053 . . 3 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) → (𝑌𝐶𝜓))
8 elmptrab2OLD.rc . . . . . 6 (𝑌𝐶𝑋𝑊)
9 elex 3185 . . . . . 6 (𝑋𝑊𝑋 ∈ V)
108, 9syl 17 . . . . 5 (𝑌𝐶𝑋 ∈ V)
1110adantr 480 . . . 4 ((𝑌𝐶𝜓) → 𝑋 ∈ V)
12 simpl 472 . . . 4 ((𝑌𝐶𝜓) → 𝑌𝐶)
13 simpr 476 . . . 4 ((𝑌𝐶𝜓) → 𝜓)
1411, 12, 133jca 1235 . . 3 ((𝑌𝐶𝜓) → (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
157, 14impbii 198 . 2 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) ↔ (𝑌𝐶𝜓))
166, 15bitri 263 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by: (None)
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