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Theorem fliftrel 5792
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 4658 . . . . 5 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
52, 3, 4syl2anc 411 . . . 4 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
6 eqid 2177 . . . 4 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
75, 6fmptd 5670 . . 3 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑅 × 𝑆))
8 frn 5374 . . 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 3201 1 (𝜑𝐹 ⊆ (𝑅 × 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wss 3129  cop 3595  cmpt 4064   × cxp 4624  ran crn 4627  wf 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224
This theorem is referenced by:  fliftcnv  5795  fliftfun  5796  fliftf  5799  qliftrel  6613
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