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Theorem fliftel 6879
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
fliftel (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑅   𝑥,𝐷   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftel
StepHypRef Expression
1 df-br 4924 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2 flift.1 . . . 4 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32eleq2i 2851 . . 3 (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
4 eqid 2772 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
5 opex 5207 . . . 4 𝐴, 𝐵⟩ ∈ V
64, 5elrnmpti 5669 . . 3 (⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
71, 3, 63bitri 289 . 2 (𝐶𝐹𝐷 ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
8 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
9 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
10 opthg2 5222 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
118, 9, 10syl2anc 576 . . 3 ((𝜑𝑥𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
1211rexbidva 3235 . 2 (𝜑 → (∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
137, 12syl5bb 275 1 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3083  cop 4441   class class class wbr 4923  cmpt 5002  ran crn 5402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-mpt 5003  df-cnv 5409  df-dm 5411  df-rn 5412
This theorem is referenced by:  fliftcnv  6881  fliftfun  6882  fliftf  6885  fliftval  6886  qliftel  8174
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