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Theorem elmptrab2 21849
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 21848 . 2 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
7 3simpc 1175 . . 3 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) → (𝑌𝐶𝜓))
8 elmptrab2.rc . . . . . 6 (𝑌𝐶𝑋𝑊)
9 elex 3413 . . . . . 6 (𝑋𝑊𝑋 ∈ V)
108, 9syl 17 . . . . 5 (𝑌𝐶𝑋 ∈ V)
1110adantr 468 . . . 4 ((𝑌𝐶𝜓) → 𝑋 ∈ V)
12 simpl 470 . . . 4 ((𝑌𝐶𝜓) → 𝑌𝐶)
13 simpr 473 . . . 4 ((𝑌𝐶𝜓) → 𝜓)
1411, 12, 133jca 1151 . . 3 ((𝑌𝐶𝜓) → (𝑋 ∈ V ∧ 𝑌𝐶𝜓))
157, 14impbii 200 . 2 ((𝑋 ∈ V ∧ 𝑌𝐶𝜓) ↔ (𝑌𝐶𝜓))
166, 15bitri 266 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑌𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  {crab 3107  Vcvv 3398  cmpt 4930  cfv 6104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fv 6112
This theorem is referenced by:  isfil  21868  isufil  21924
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