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Theorem fliftel1 6557
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 4930 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 eqid 2621 . . . . . 6 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32elrnmpt1 5372 . . . . 5 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3mpan2 707 . . . 4 (𝑥𝑋 → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
54adantl 482 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
6 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6syl6eleqr 2711 . 2 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8 df-br 4652 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
97, 8sylibr 224 1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cop 4181   class class class wbr 4651  cmpt 4727  ran crn 5113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-mpt 4728  df-cnv 5120  df-dm 5122  df-rn 5123
This theorem is referenced by:  fliftfun  6559  qliftel1  7828
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