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Theorem fliftel1 5461
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
fliftel1 ((𝜑𝑥𝑋) → 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 ((𝜑𝑥𝑋) → 𝐴𝑅)
2 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
3 opexg 3991 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ V)
41, 2, 3syl2anc 397 . . . 4 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ V)
5 eqid 2056 . . . . . 6 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
65elrnmpt1 4612 . . . . 5 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
76adantll 453 . . . 4 (((𝜑𝑥𝑋) ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
84, 7mpdan 406 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
9 flift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
108, 9syl6eleqr 2147 . 2 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ 𝐹)
11 df-br 3792 . 2 (𝐴𝐹𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹)
1210, 11sylibr 141 1 ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cop 3405   class class class wbr 3791  cmpt 3845  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-mpt 3847  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  fliftfun  5463  qliftel1  6217
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