Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fliftel1 Structured version   Visualization version   GIF version

Theorem fliftel1 6600
 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 4962 . . . . 5 𝐴, 𝐵⟩ ∈ V
2 eqid 2651 . . . . . 6 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32elrnmpt1 5406 . . . . 5 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3mpan2 707 . . . 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, 6syl6eleqr 2741 . 2 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8 df-br 4686 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
97, 8sylibr 224 1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154 This theorem is referenced by:  fliftfun  6602  qliftel1  7874
 Copyright terms: Public domain W3C validator