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Theorem genpss 11044
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 11040 . . 3 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 elprnq 11031 . . . . . . . 8 ((𝐴P𝑔𝐴) → 𝑔Q)
54ex 412 . . . . . . 7 (𝐴P → (𝑔𝐴𝑔Q))
6 elprnq 11031 . . . . . . . 8 ((𝐵P𝐵) → Q)
76ex 412 . . . . . . 7 (𝐵P → (𝐵Q))
85, 7im2anan9 620 . . . . . 6 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔QQ)))
92caovcl 7627 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
108, 9syl6 35 . . . . 5 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔𝐺) ∈ Q))
11 eleq1a 2836 . . . . 5 ((𝑔𝐺) ∈ Q → (𝑓 = (𝑔𝐺) → 𝑓Q))
1210, 11syl6 35 . . . 4 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → 𝑓Q)))
1312rexlimdvv 3212 . . 3 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → 𝑓Q))
143, 13sylbid 240 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓Q))
1514ssrdv 3989 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  wss 3951  (class class class)co 7431  cmpo 7433  Qcnq 10892  Pcnp 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-ni 10912  df-nq 10952  df-np 11021
This theorem is referenced by:  genpcl  11048
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