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Theorem genipdm 6498
Description: Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
Assertion
Ref Expression
genipdm 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 genipdm
StepHypRef Expression
1 genpelvl.1 . 2 𝐹 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
2 nqex 6347 . . . 4 Q V
32rabex 3892 . . 3 {x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))} V
42rabex 3892 . . 3 {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))} V
53, 4opex 3957 . 2 ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩ V
61, 5dmmpt2 5773 1 dom 𝐹 = (P × P)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wrex 2301  {crab 2304  cop 3370   × cxp 4286  dom cdm 4288  cfv 4845  (class class class)co 5455  cmpt2 5457  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332
This theorem is referenced by:  dmplp  6523  dmmp  6524
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