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Theorem fliftel1 5377
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
fliftel1 ((φ x 𝑋) → A𝐹B)
Distinct variable groups:   x,𝑅   φ,x   x,𝑋   x,𝑆
Allowed substitution hints:   A(x)   B(x)   𝐹(x)

Proof of Theorem fliftel1
StepHypRef Expression
1 flift.2 . . . . 5 ((φ x 𝑋) → A 𝑅)
2 flift.3 . . . . 5 ((φ x 𝑋) → B 𝑆)
3 opexg 3955 . . . . 5 ((A 𝑅 B 𝑆) → ⟨A, B V)
41, 2, 3syl2anc 391 . . . 4 ((φ x 𝑋) → ⟨A, B V)
5 eqid 2037 . . . . . 6 (x 𝑋 ↦ ⟨A, B⟩) = (x 𝑋 ↦ ⟨A, B⟩)
65elrnmpt1 4528 . . . . 5 ((x 𝑋 A, B V) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
76adantll 445 . . . 4 (((φ x 𝑋) A, B V) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
84, 7mpdan 398 . . 3 ((φ x 𝑋) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
9 flift.1 . . 3 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
108, 9syl6eleqr 2128 . 2 ((φ x 𝑋) → ⟨A, B 𝐹)
11 df-br 3756 . 2 (A𝐹B ↔ ⟨A, B 𝐹)
1210, 11sylibr 137 1 ((φ x 𝑋) → A𝐹B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  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-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  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:  fliftfun  5379  qliftel1  6123
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