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Theorem fliftel 6436
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 4578 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2 flift.1 . . . 4 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32eleq2i 2679 . . 3 (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
4 eqid 2609 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
5 opex 4852 . . . 4 𝐴, 𝐵⟩ ∈ V
64, 5elrnmpti 5283 . . 3 (⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
71, 3, 63bitri 284 . 2 (𝐶𝐹𝐷 ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩)
8 flift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑅)
9 flift.3 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑆)
10 opthg2 4867 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
118, 9, 10syl2anc 690 . . 3 ((𝜑𝑥𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
1211rexbidva 3030 . 2 (𝜑 → (∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
137, 12syl5bb 270 1 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  cop 4130   class class class wbr 4577  cmpt 4637  ran crn 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-cnv 5035  df-dm 5037  df-rn 5038
This theorem is referenced by:  fliftcnv  6438  fliftfun  6439  fliftf  6442  fliftval  6443  qliftel  7694
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