ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fliftel GIF version

Theorem fliftel 5512
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 3812 . . . 4 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32eleq2i 2149 . . . 4 (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3bitri 182 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
5 flift.2 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑅)
6 flift.3 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵𝑆)
7 opexg 4019 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ V)
85, 6, 7syl2anc 403 . . . . 5 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
98ralrimiva 2440 . . . 4 (𝜑 → ∀𝑥𝑋𝐴, 𝐵⟩ ∈ V)
10 eqid 2083 . . . . 5 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1110elrnmptg 4645 . . . 4 (∀𝑥𝑋𝐴, 𝐵⟩ ∈ V → (⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩))
129, 11syl 14 . . 3 (𝜑 → (⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩))
134, 12syl5bb 190 . 2 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩))
14 opthg2 4030 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
155, 6, 14syl2anc 403 . . 3 ((𝜑𝑥𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
1615rexbidva 2371 . 2 (𝜑 → (∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1713, 16bitrd 186 1 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353  wrex 2354  Vcvv 2612  cop 3425   class class class wbr 3811  cmpt 3865  ran crn 4402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-mpt 3867  df-cnv 4409  df-dm 4411  df-rn 4412
This theorem is referenced by:  fliftcnv  5514  fliftfun  5515  fliftf  5518  fliftval  5519  qliftel  6302
  Copyright terms: Public domain W3C validator