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Theorem genpss 10936
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 10932 . . 3 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 elprnq 10923 . . . . . . . 8 ((𝐴P𝑔𝐴) → 𝑔Q)
54ex 413 . . . . . . 7 (𝐴P → (𝑔𝐴𝑔Q))
6 elprnq 10923 . . . . . . . 8 ((𝐵P𝐵) → Q)
76ex 413 . . . . . . 7 (𝐵P → (𝐵Q))
85, 7im2anan9 620 . . . . . 6 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔QQ)))
92caovcl 7544 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
108, 9syl6 35 . . . . 5 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔𝐺) ∈ Q))
11 eleq1a 2833 . . . . 5 ((𝑔𝐺) ∈ Q → (𝑓 = (𝑔𝐺) → 𝑓Q))
1210, 11syl6 35 . . . 4 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → 𝑓Q)))
1312rexlimdvv 3202 . . 3 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → 𝑓Q))
143, 13sylbid 239 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓Q))
1514ssrdv 3948 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2713  wrex 3071  wss 3908  (class class class)co 7353  cmpo 7355  Qcnq 10784  Pcnp 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-inf2 9573
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-ni 10804  df-nq 10844  df-np 10913
This theorem is referenced by:  genpcl  10940
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