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Theorem genpelv 10426
 Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (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
genpelv ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑔,   𝑔,𝐹   𝐶,𝑔,
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpelv
Dummy variable 𝑓 is 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, 2genpv 10425 . . 3 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
43eleq2d 2875 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}))
5 id 22 . . . . . 6 (𝐶 = (𝑔𝐺) → 𝐶 = (𝑔𝐺))
6 ovex 7175 . . . . . 6 (𝑔𝐺) ∈ V
75, 6eqeltrdi 2898 . . . . 5 (𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
87rexlimivw 3241 . . . 4 (∃𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
98rexlimivw 3241 . . 3 (∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
10 eqeq1 2802 . . . 4 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
11102rexbidv 3259 . . 3 (𝑓 = 𝐶 → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
129, 11elab3 3622 . 2 (𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)} ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺))
134, 12syl6bb 290 1 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441  (class class class)co 7142   ∈ cmpo 7144  Qcnq 10278  Pcnp 10285 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-inf2 9103 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7571  df-ni 10298  df-nq 10338  df-np 10407 This theorem is referenced by:  genpprecl  10427  genpss  10430  genpnnp  10431  genpcd  10432  genpnmax  10433  genpass  10435  distrlem1pr  10451  distrlem5pr  10453  1idpr  10455  ltexprlem6  10467  reclem3pr  10475  reclem4pr  10476
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