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Theorem elintfv 32176
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 4673 . 2 (𝑋 (𝐹𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧))
3 ssel2 3793 . . . . . . . . . 10 ((𝐵𝐴𝑦𝐵) → 𝑦𝐴)
4 fnbrfvb 6460 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑦𝐴) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
53, 4sylan2 587 . . . . . . . . 9 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴𝑦𝐵)) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
65anassrs 460 . . . . . . . 8 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑦𝐵) → ((𝐹𝑦) = 𝑧𝑦𝐹𝑧))
76rexbidva 3230 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑦𝐵 (𝐹𝑦) = 𝑧 ↔ ∃𝑦𝐵 𝑦𝐹𝑧))
8 vex 3388 . . . . . . . 8 𝑧 ∈ V
98elima 5688 . . . . . . 7 (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 𝑦𝐹𝑧)
107, 9syl6rbbr 282 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑧 ∈ (𝐹𝐵) ↔ ∃𝑦𝐵 (𝐹𝑦) = 𝑧))
1110imbi1d 333 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧)))
12 r19.23v 3204 . . . . 5 (∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (∃𝑦𝐵 (𝐹𝑦) = 𝑧𝑋𝑧))
1311, 12syl6bbr 281 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
1413albidv 2016 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧)))
15 ralcom4 3412 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧))
16 eqcom 2806 . . . . . . . 8 ((𝐹𝑦) = 𝑧𝑧 = (𝐹𝑦))
1716imbi1i 341 . . . . . . 7 (((𝐹𝑦) = 𝑧𝑋𝑧) ↔ (𝑧 = (𝐹𝑦) → 𝑋𝑧))
1817albii 1915 . . . . . 6 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧))
19 fvex 6424 . . . . . . 7 (𝐹𝑦) ∈ V
20 eleq2 2867 . . . . . . 7 (𝑧 = (𝐹𝑦) → (𝑋𝑧𝑋 ∈ (𝐹𝑦)))
2119, 20ceqsalv 3421 . . . . . 6 (∀𝑧(𝑧 = (𝐹𝑦) → 𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
2218, 21bitri 267 . . . . 5 (∀𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ 𝑋 ∈ (𝐹𝑦))
2322ralbii 3161 . . . 4 (∀𝑦𝐵𝑧((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
2415, 23bitr3i 269 . . 3 (∀𝑧𝑦𝐵 ((𝐹𝑦) = 𝑧𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦))
2514, 24syl6bb 279 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑧(𝑧 ∈ (𝐹𝐵) → 𝑋𝑧) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
262, 25syl5bb 275 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  wral 3089  wrex 3090  Vcvv 3385  wss 3769   cint 4667   class class class wbr 4843  cima 5315   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by: (None)
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