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Theorem fliftel1 6788
Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2 ((𝜑𝑥𝑋) → 𝐴𝑅)
flift.3 ((𝜑𝑥𝑋) → 𝐵𝑆)
Assertion
Ref Expression
fliftel1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftel1
StepHypRef Expression
1 opex 5123 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 eqid 2799 . . . . . 6 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32elrnmpt1 5578 . . . . 5 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3mpan2 683 . . . 4 (𝑥𝑋 → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
54adantl 474 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
6 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6syl6eleqr 2889 . 2 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8 df-br 4844 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
97, 8sylibr 226 1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374   class class class wbr 4843  cmpt 4922  ran crn 5313
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-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  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-br 4844  df-opab 4906  df-mpt 4923  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  fliftfun  6790  qliftel1  8069
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