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Theorem fliftrel 5375
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 (x 𝑋 ↦ ⟨A, B⟩)
flift.2 ((φ x 𝑋) → A 𝑅)
flift.3 ((φ x 𝑋) → B 𝑆)
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 (x 𝑋 ↦ ⟨A, B⟩)
2 flift.2 . . . . 5 ((φ x 𝑋) → A 𝑅)
3 flift.3 . . . . 5 ((φ x 𝑋) → B 𝑆)
4 opelxpi 4319 . . . . 5 ((A 𝑅 B 𝑆) → ⟨A, B (𝑅 × 𝑆))
52, 3, 4syl2anc 391 . . . 4 ((φ x 𝑋) → ⟨A, B (𝑅 × 𝑆))
6 eqid 2037 . . . 4 (x 𝑋 ↦ ⟨A, B⟩) = (x 𝑋 ↦ ⟨A, B⟩)
75, 6fmptd 5265 . . 3 (φ → (x 𝑋 ↦ ⟨A, B⟩):𝑋⟶(𝑅 × 𝑆))
8 frn 4995 . . 3 ((x 𝑋 ↦ ⟨A, B⟩):𝑋⟶(𝑅 × 𝑆) → ran (x 𝑋 ↦ ⟨A, B⟩) ⊆ (𝑅 × 𝑆))
97, 8syl 14 . 2 (φ → ran (x 𝑋 ↦ ⟨A, B⟩) ⊆ (𝑅 × 𝑆))
101, 9syl5eqss 2983 1 (φ𝐹 ⊆ (𝑅 × 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wss 2911  cop 3370  cmpt 3809   × cxp 4286  ran crn 4289  wf 4841
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by:  fliftcnv  5378  fliftfun  5379  fliftf  5382  qliftrel  6121
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