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Theorem fliftrel 5702
 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)
4 opelxpi 4580 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
52, 3, 4syl2anc 409 . . . 4 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
6 eqid 2140 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6fmptd 5583 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆))
8 frn 5290 . . 3 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆) → ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ⊆ (𝑅 × 𝑆))
97, 8syl 14 . 2 (𝜑 → ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ⊆ (𝑅 × 𝑆))
101, 9eqsstrid 3149 1 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481   ⊆ wss 3077  ⟨cop 3536   ↦ cmpt 3998   × cxp 4546  ran crn 4549  ⟶wf 5128 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fv 5140 This theorem is referenced by:  fliftcnv  5705  fliftfun  5706  fliftf  5709  qliftrel  6517
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