ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fliftel1 GIF version

Theorem fliftel1 5380
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 3958 . . . . 5 ((A 𝑅 B 𝑆) → ⟨A, B V)
41, 2, 3syl2anc 391 . . . 4 ((φ x 𝑋) → ⟨A, B V)
5 eqid 2040 . . . . . 6 (x 𝑋 ↦ ⟨A, B⟩) = (x 𝑋 ↦ ⟨A, B⟩)
65elrnmpt1 4531 . . . . 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 2131 . 2 ((φ x 𝑋) → ⟨A, B 𝐹)
11 df-br 3759 . 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 1243   wcel 1393  Vcvv 2554  cop 3373   class class class wbr 3758  cmpt 3812  ran crn 4292
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-mpt 3814  df-cnv 4299  df-dm 4301  df-rn 4302
This theorem is referenced by:  fliftfun  5382  qliftel1  6126
  Copyright terms: Public domain W3C validator