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Theorem elmptrab2 23817
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 23816 . 2 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
7 3simpc 1147 . . 3 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) → (𝑌𝐶𝜓))
8 elmptrab2.rc . . . . . 6 (𝑌𝐶𝑋𝑊)
98elexd 3485 . . . . 5 (𝑌𝐶𝑋 ∈ V)
109adantr 479 . . . 4 ((𝑌𝐶𝜓) → 𝑋 ∈ V)
11 simpl 481 . . . 4 ((𝑌𝐶𝜓) → 𝑌𝐶)
12 simpr 483 . . . 4 ((𝑌𝐶𝜓) → 𝜓)
1310, 11, 123jca 1125 . . 3 ((𝑌𝐶𝜓) → (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
147, 13impbii 208 . 2 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) ↔ (𝑌𝐶𝜓))
156, 14bitri 274 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cmpt 5226  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  isfil  23836  isufil  23892
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