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Theorem elintfv 34731
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 4956 . 2 (𝑋 (𝐹𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧))
3 fvelimab 6964 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑧))
43imbi1d 341 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧)))
5 r19.23v 3182 . . . . 5 (∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧))
64, 5bitr4di 288 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
76albidv 1923 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
8 ralcom4 3283 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧))
9 eqcom 2739 . . . . . . . 8 ((𝐹𝑦) = 𝑧𝑧 = (𝐹𝑦))
109imbi1i 349 . . . . . . 7 (((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (𝑧 = (𝐹𝑦) → 𝑋𝑧))
1110albii 1821 . . . . . 6 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧))
12 fvex 6904 . . . . . . 7 (𝐹𝑦) ∈ V
13 eleq2 2822 . . . . . . 7 (𝑧 = (𝐹𝑦) → (𝑋𝑧𝑋 ∈ (𝐹𝑦)))
1412, 13ceqsalv 3511 . . . . . 6 (∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1511, 14bitri 274 . . . . 5 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
1615ralbii 3093 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
178, 16bitr3i 276 . . 3 (∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
187, 17bitrdi 286 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
192, 18bitrid 282 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  wss 3948   cint 4950  cima 5679   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by: (None)
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