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Theorem fliftrel 7117
Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (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
fliftrel (𝜑𝐹 ⊆ (𝑅 × 𝑆))
Distinct variable groups:   𝑥,𝑅   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftrel
StepHypRef Expression
1 flift.1 . 2 𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
2 flift.2 . . . . 5 ((𝜑𝑥𝑋) → 𝐴𝑅)
3 flift.3 . . . . 5 ((𝜑𝑥𝑋) → 𝐵𝑆)
42, 3opelxpd 5589 . . . 4 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
54fmpttd 6932 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆))
65frnd 6553 . 2 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ⊆ (𝑅 × 𝑆))
71, 6eqsstrid 3949 1 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wss 3866  cop 4547  cmpt 5135   × cxp 5549  ran crn 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-f 6384
This theorem is referenced by:  fliftcnv  7120  fliftfun  7121  fliftf  7124  qliftrel  8481  fmucndlem  23188  pi1xfrcnv  23954
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