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Theorem genpelv 9766
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 9765 . . 3 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
43eleq2d 2684 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}))
5 id 22 . . . . . 6 (𝐶 = (𝑔𝐺) → 𝐶 = (𝑔𝐺))
6 ovex 6632 . . . . . 6 (𝑔𝐺) ∈ V
75, 6syl6eqel 2706 . . . . 5 (𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
87rexlimivw 3022 . . . 4 (∃𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
98rexlimivw 3022 . . 3 (∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
10 eqeq1 2625 . . . 4 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
11102rexbidv 3050 . . 3 (𝑓 = 𝐶 → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
129, 11elab3 3341 . 2 (𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)} ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺))
134, 12syl6bb 276 1 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  (class class class)co 6604  cmpt2 6606  Qcnq 9618  Pcnp 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-ni 9638  df-nq 9678  df-np 9747
This theorem is referenced by:  genpprecl  9767  genpss  9770  genpnnp  9771  genpcd  9772  genpnmax  9773  genpass  9775  distrlem1pr  9791  distrlem5pr  9793  1idpr  9795  ltexprlem6  9807  reclem3pr  9815  reclem4pr  9816
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