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Theorem genpss 10161
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 10157 . . 3 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 elprnq 10148 . . . . . . . 8 ((𝐴P𝑔𝐴) → 𝑔Q)
54ex 403 . . . . . . 7 (𝐴P → (𝑔𝐴𝑔Q))
6 elprnq 10148 . . . . . . . 8 ((𝐵P𝐵) → Q)
76ex 403 . . . . . . 7 (𝐵P → (𝐵Q))
85, 7im2anan9 613 . . . . . 6 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔QQ)))
92caovcl 7105 . . . . . 6 ((𝑔QQ) → (𝑔𝐺) ∈ Q)
108, 9syl6 35 . . . . 5 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑔𝐺) ∈ Q))
11 eleq1a 2854 . . . . 5 ((𝑔𝐺) ∈ Q → (𝑓 = (𝑔𝐺) → 𝑓Q))
1210, 11syl6 35 . . . 4 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → 𝑓Q)))
1312rexlimdvv 3220 . . 3 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → 𝑓Q))
143, 13sylbid 232 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑓Q))
1514ssrdv 3827 1 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {cab 2763  wrex 3091  wss 3792  (class class class)co 6922  cmpt2 6924  Qcnq 10009  Pcnp 10016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-ni 10029  df-nq 10069  df-np 10138
This theorem is referenced by:  genpcl  10165
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