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Theorem xpsfrn 17279
Description: A short expression for the indexed cartesian product on two indices. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
Assertion
Ref Expression
xpsfrn ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Distinct variable groups:   𝐴,𝑘,𝑥,𝑦   𝐵,𝑘,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑘)

Proof of Theorem xpsfrn
StepHypRef Expression
1 xpsff1o.f . . 3 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
21xpsff1o 17278 . 2 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
3 f1ofo 6723 . 2 (𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
4 forn 6691 . 2 (𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
52, 3, 4mp2b 10 1 ran 𝐹 = X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  c0 4256  ifcif 4459  {cpr 4563  cop 4567   × cxp 5587  ran crn 5590  ontowfo 6431  1-1-ontowf1o 6432  cmpo 7277  1oc1o 8290  2oc2o 8291  Xcixp 8685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-2o 8298  df-ixp 8686  df-en 8734  df-fin 8737
This theorem is referenced by:  xpsrnbas  17282
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