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Theorem fliftel 5376
Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
flift.2 ((φ x 𝑋) → A 𝑅)
flift.3 ((φ x 𝑋) → B 𝑆)
Assertion
Ref Expression
fliftel (φ → (𝐶𝐹𝐷x 𝑋 (𝐶 = A 𝐷 = 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 3756 . . . 4 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷 𝐹)
2 flift.1 . . . . 5 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
32eleq2i 2101 . . . 4 (⟨𝐶, 𝐷 𝐹 ↔ ⟨𝐶, 𝐷 ran (x 𝑋 ↦ ⟨A, B⟩))
41, 3bitri 173 . . 3 (𝐶𝐹𝐷 ↔ ⟨𝐶, 𝐷 ran (x 𝑋 ↦ ⟨A, B⟩))
5 flift.2 . . . . . 6 ((φ x 𝑋) → A 𝑅)
6 flift.3 . . . . . 6 ((φ x 𝑋) → B 𝑆)
7 elex 2560 . . . . . . 7 (A 𝑅A V)
8 elex 2560 . . . . . . 7 (B 𝑆B V)
9 opexgOLD 3956 . . . . . . 7 ((A V B V) → ⟨A, B V)
107, 8, 9syl2an 273 . . . . . 6 ((A 𝑅 B 𝑆) → ⟨A, B V)
115, 6, 10syl2anc 391 . . . . 5 ((φ x 𝑋) → ⟨A, B V)
1211ralrimiva 2386 . . . 4 (φx 𝑋A, B V)
13 eqid 2037 . . . . 5 (x 𝑋 ↦ ⟨A, B⟩) = (x 𝑋 ↦ ⟨A, B⟩)
1413elrnmptg 4529 . . . 4 (x 𝑋A, B V → (⟨𝐶, 𝐷 ran (x 𝑋 ↦ ⟨A, B⟩) ↔ x 𝑋𝐶, 𝐷⟩ = ⟨A, B⟩))
1512, 14syl 14 . . 3 (φ → (⟨𝐶, 𝐷 ran (x 𝑋 ↦ ⟨A, B⟩) ↔ x 𝑋𝐶, 𝐷⟩ = ⟨A, B⟩))
164, 15syl5bb 181 . 2 (φ → (𝐶𝐹𝐷x 𝑋𝐶, 𝐷⟩ = ⟨A, B⟩))
17 opthg2 3967 . . . 4 ((A 𝑅 B 𝑆) → (⟨𝐶, 𝐷⟩ = ⟨A, B⟩ ↔ (𝐶 = A 𝐷 = B)))
185, 6, 17syl2anc 391 . . 3 ((φ x 𝑋) → (⟨𝐶, 𝐷⟩ = ⟨A, B⟩ ↔ (𝐶 = A 𝐷 = B)))
1918rexbidva 2317 . 2 (φ → (x 𝑋𝐶, 𝐷⟩ = ⟨A, B⟩ ↔ x 𝑋 (𝐶 = A 𝐷 = B)))
2016, 19bitrd 177 1 (φ → (𝐶𝐹𝐷x 𝑋 (𝐶 = A 𝐷 = B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  cop 3370   class class class wbr 3755  cmpt 3809  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  fliftcnv  5378  fliftfun  5379  fliftf  5382  fliftval  5383  qliftel  6122
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