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Theorem fliftel 6722
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 4805 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2 flift.1 . . . 4 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32eleq2i 2831 . . 3 (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
4 eqid 2760 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
5 opex 5081 . . . 4 𝐴, 𝐵⟩ ∈ V
64, 5elrnmpti 5531 . . 3 (⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
71, 3, 63bitri 286 . 2 (𝐶𝐹𝐷 ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
8 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
9 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
10 opthg2 5096 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
118, 9, 10syl2anc 696 . . 3 ((𝜑𝑥𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
1211rexbidva 3187 . 2 (𝜑 → (∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
137, 12syl5bb 272 1 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  cop 4327   class class class wbr 4804  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  fliftcnv  6724  fliftfun  6725  fliftf  6728  fliftval  6729  qliftel  7997
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