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Theorem fliftrel 5661
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 4541 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
52, 3, 4syl2anc 408 . . . 4 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
6 eqid 2117 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6fmptd 5542 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆))
8 frn 5251 . . 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 3113 1 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  wss 3041  cop 3500  cmpt 3959   × cxp 4507  ran crn 4510  wf 5089
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  fliftcnv  5664  fliftfun  5665  fliftf  5668  qliftrel  6476
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