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Theorem elmptrab2 22430
Description: Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)
Hypotheses
Ref Expression
elmptrab2.f 𝐹 = (𝑥 ∈ V ↦ {𝑦𝐵𝜑})
elmptrab2.s1 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
elmptrab2.s2 (𝑥 = 𝑋𝐵 = 𝐶)
elmptrab2.ex 𝐵 ∈ V
elmptrab2.rc (𝑌𝐶𝑋𝑊)
Assertion
Ref Expression
elmptrab2 (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐶,𝑦   𝑥,𝑊,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem elmptrab2
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 elmptrab2.ex . . . 4 𝐵 ∈ V
54a1i 11 . . 3 (𝑥 ∈ V → 𝐵 ∈ V)
61, 2, 3, 5elmptrab 22429 . 2 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
7 3simpc 1146 . . 3 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) → (𝑌𝐶𝜓))
8 elmptrab2.rc . . . . . 6 (𝑌𝐶𝑋𝑊)
98elexd 3514 . . . . 5 (𝑌𝐶𝑋 ∈ V)
109adantr 483 . . . 4 ((𝑌𝐶𝜓) → 𝑋 ∈ V)
11 simpl 485 . . . 4 ((𝑌𝐶𝜓) → 𝑌𝐶)
12 simpr 487 . . . 4 ((𝑌𝐶𝜓) → 𝜓)
1310, 11, 123jca 1124 . . 3 ((𝑌𝐶𝜓) → (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
147, 13impbii 211 . 2 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) ↔ (𝑌𝐶𝜓))
156, 14bitri 277 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  {crab 3142  Vcvv 3494  cmpt 5138  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fv 6357
This theorem is referenced by:  isfil  22449  isufil  22505
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