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Theorem fliftel 5662
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 3900 . . . 4 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐹)
2 flift.1 . . . . 5 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
32eleq2i 2184 . . . 4 (⟨𝐶, 𝐷⟩ ∈ 𝐹 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
41, 3bitri 183 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
5 flift.2 . . . . . 6 ((𝜑𝑥𝑋) → 𝐴𝑅)
6 flift.3 . . . . . 6 ((𝜑𝑥𝑋) → 𝐵𝑆)
7 opexg 4120 . . . . . 6 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ V)
85, 6, 7syl2anc 408 . . . . 5 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
98ralrimiva 2482 . . . 4 (𝜑 → ∀𝑥𝑋𝐴, 𝐵⟩ ∈ V)
10 eqid 2117 . . . . 5 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1110elrnmptg 4761 . . . 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 191 . 2 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩))
14 opthg2 4131 . . . 4 ((𝐴𝑅𝐵𝑆) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
155, 6, 14syl2anc 408 . . 3 ((𝜑𝑥𝑋) → (⟨𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝐶 = 𝐴𝐷 = 𝐵)))
1615rexbidva 2411 . 2 (𝜑 → (∃𝑥𝑋𝐶, 𝐷⟩ = ⟨𝐴, 𝐵⟩ ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
1713, 16bitrd 187 1 (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  wral 2393  wrex 2394  Vcvv 2660  cop 3500   class class class wbr 3899  cmpt 3959  ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  fliftcnv  5664  fliftfun  5665  fliftf  5668  fliftval  5669  qliftel  6477
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