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Theorem genpss 11028
Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpss ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpss
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2 genp.2 . . . 4 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpelv 11024 . . 3 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 elprnq 11015 . . . . . . . 8 ((𝐴P𝑔𝐴) → 𝑔Q)
54ex 412 . . . . . . 7 (𝐴P → (𝑔𝐴𝑔Q))
6 elprnq 11015 . . . . . . . 8 ((𝐵P𝐵) → Q)
76ex 412 . . . . . . 7 (𝐵P → (𝐵Q))
85, 7im2anan9 619 . . . . . 6 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔QQ)))
92caovcl 7615 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
108, 9syl6 35 . . . . 5 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔𝐺) ∈ Q))
11 eleq1a 2824 . . . . 5 ((𝑔𝐺) ∈ Q → (𝑓 = (𝑔𝐺) → 𝑓Q))
1210, 11syl6 35 . . . 4 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → 𝑓Q)))
1312rexlimdvv 3207 . . 3 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → 𝑓Q))
143, 13sylbid 239 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓Q))
1514ssrdv 3986 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  wss 3947  (class class class)co 7420  cmpo 7422  Qcnq 10876  Pcnp 10883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-ni 10896  df-nq 10936  df-np 11005
This theorem is referenced by:  genpcl  11032
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