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Theorem fliftel1 5359
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 3938 . . . . 5 ((A 𝑅 B 𝑆) → ⟨A, B V)
41, 2, 3syl2anc 393 . . . 4 ((φ x 𝑋) → ⟨A, B V)
5 eqid 2022 . . . . . 6 (x 𝑋 ↦ ⟨A, B⟩) = (x 𝑋 ↦ ⟨A, B⟩)
65elrnmpt1 4512 . . . . 5 ((x 𝑋 A, B V) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
76adantll 448 . . . 4 (((φ x 𝑋) A, B V) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
84, 7mpdan 400 . . 3 ((φ x 𝑋) → ⟨A, B ran (x 𝑋 ↦ ⟨A, B⟩))
9 flift.1 . . 3 𝐹 = ran (x 𝑋 ↦ ⟨A, B⟩)
108, 9syl6eleqr 2113 . 2 ((φ x 𝑋) → ⟨A, B 𝐹)
11 df-br 3739 . 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 1228   wcel 1374  Vcvv 2535  cop 3353   class class class wbr 3738  cmpt 3792  ran crn 4273
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-mpt 3794  df-cnv 4280  df-dm 4282  df-rn 4283
This theorem is referenced by:  fliftfun  5361  qliftel1  6098
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