Theorem fliftel1 7302

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 5457	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		eqid 2726	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
3	2	elrnmpt1 5950	. . . . 5 ⊢ ((𝑥 ∈ 𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
4	1, 3	mpan2 688	. . . 4 ⊢ (𝑥 ∈ 𝑋 → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩))
5	4	adantl 481	. . 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 2838	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
8		df-br 5142	. 2 ⊢ (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
9	7, 8	sylibr 233	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629   class class class wbr 5141   ↦ cmpt 5224  ran crn 5670

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-dm 5679  df-rn 5680

This theorem is referenced by:  fliftfun  7304  qliftel1  8794