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Theorem fliftel1 7299
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 5454 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 eqid 2724 . . . . . 6 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32elrnmpt1 5947 . . . . 5 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3mpan2 688 . . . 4 (𝑥𝑋 → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
54adantl 481 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
6 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6eleqtrrdi 2836 . 2 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8 df-br 5139 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
97, 8sylibr 233 1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cop 4626   class class class wbr 5138  cmpt 5221  ran crn 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-mpt 5222  df-cnv 5674  df-dm 5676  df-rn 5677
This theorem is referenced by:  fliftfun  7301  qliftel1  8791
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