Theorem fliftel1 7057

Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)

Assertion

Ref	Expression
fliftel1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 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

Step	Hyp	Ref	Expression
1		opex 5324	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		eqid 2758	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3	2	elrnmpt1 5799	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
4	1, 3	mpan2 690	. . . 4 ⊢ (𝑥 ∈ 𝑋 → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
5	4	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
6		flift.1	. . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
7	5, 6	eleqtrrdi 2863	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8		df-br 5033	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
9	7, 8	sylibr 237	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112  ran crn 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rex 3076  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-mpt 5113  df-cnv 5532  df-dm 5534  df-rn 5535

This theorem is referenced by:  fliftfun  7059  qliftel1  8391