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Theorem elintfv 33002
Description: Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
Hypothesis
Ref Expression
elintfv.1 𝑋 ∈ V
Assertion
Ref Expression
elintfv ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐹   𝑦,𝑋

Proof of Theorem elintfv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elintfv.1 . . 3 𝑋 ∈ V
21elint 4874 . 2 (𝑋 (𝐹𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧))
3 fvelimab 6731 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑧))
43imbi1d 344 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧)))
5 r19.23v 3279 . . . . 5 (∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧))
64, 5syl6bbr 291 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
76albidv 1917 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
8 ralcom4 3235 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧))
9 eqcom 2828 . . . . . . . 8 ((𝐹𝑦) = 𝑧𝑧 = (𝐹𝑦))
109imbi1i 352 . . . . . . 7 (((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (𝑧 = (𝐹𝑦) → 𝑋𝑧))
1110albii 1816 . . . . . 6 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧))
12 fvex 6677 . . . . . . 7 (𝐹𝑦) ∈ V
13 eleq2 2901 . . . . . . 7 (𝑧 = (𝐹𝑦) → (𝑋𝑧𝑋 ∈ (𝐹𝑦)))
1412, 13ceqsalv 3532 . . . . . 6 (∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1511, 14bitri 277 . . . . 5 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1615ralbii 3165 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
178, 16bitr3i 279 . . 3 (∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
187, 17syl6bb 289 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
192, 18syl5bb 285 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1531   = wceq 1533  wcel 2110  wral 3138  wrex 3139  Vcvv 3494  wss 3935   cint 4868  cima 5552   Fn wfn 6344  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-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-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-int 4869  df-br 5059  df-opab 5121  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-fn 6352  df-fv 6357
This theorem is referenced by: (None)
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