MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  genpdm Structured version   Visualization version   GIF version

Theorem genpdm 9703
Description: Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpdm dom 𝐹 = (P × P)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝐺
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpdm
StepHypRef Expression
1 elprnq 9692 . . . . . . . 8 ((𝑤P𝑦𝑤) → 𝑦Q)
2 elprnq 9692 . . . . . . . 8 ((𝑣P𝑧𝑣) → 𝑧Q)
3 genp.2 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
4 eleq1 2676 . . . . . . . . 9 (𝑥 = (𝑦𝐺𝑧) → (𝑥Q ↔ (𝑦𝐺𝑧) ∈ Q))
53, 4syl5ibrcom 236 . . . . . . . 8 ((𝑦Q𝑧Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
61, 2, 5syl2an 493 . . . . . . 7 (((𝑤P𝑦𝑤) ∧ (𝑣P𝑧𝑣)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
76an4s 865 . . . . . 6 (((𝑤P𝑣P) ∧ (𝑦𝑤𝑧𝑣)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
87rexlimdvva 3020 . . . . 5 ((𝑤P𝑣P) → (∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧) → 𝑥Q))
98abssdv 3639 . . . 4 ((𝑤P𝑣P) → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
10 nqex 9624 . . . 4 Q ∈ V
11 ssexg 4732 . . . 4 (({𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ⊆ QQ ∈ V) → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ∈ V)
129, 10, 11sylancl 693 . . 3 ((𝑤P𝑣P) → {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ∈ V)
1312rgen2a 2960 . 2 𝑤P𝑣P {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ∈ V
14 genp.1 . . 3 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
1514fnmpt2 7127 . 2 (∀𝑤P𝑣P {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)} ∈ V → 𝐹 Fn (P × P))
16 fndm 5904 . 2 (𝐹 Fn (P × P) → dom 𝐹 = (P × P))
1713, 15, 16mp2b 10 1 dom 𝐹 = (P × P)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  wss 3540   × cxp 5036  dom cdm 5038   Fn wfn 5799  (class class class)co 6549  cmpt2 6551  Qcnq 9553  Pcnp 9560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-ni 9573  df-nq 9613  df-np 9682
This theorem is referenced by:  dmplp  9713  dmmp  9714
  Copyright terms: Public domain W3C validator