Theorem genpdm 3899

Description: Domain of general operation on positive reals.

Hypothesis

Ref	Expression
genp.1	⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}

Assertion

Ref	Expression
genpdm	⊢ dom F = (P × P)

Distinct variable group(s):   x,y,z,w,v,u,G

Proof of Theorem genpdm

Step	Hyp	Ref	Expression
1		dmoprab 3031	. 2 ⊢ dom {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})} = {⟨w, v⟩∣∃u((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
2		genp.1	. . 3 ⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
3	2	dmeqi 2532	. 2 ⊢ dom F = dom {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
4		df-xp 2424	. . 3 ⊢ (P × P) = {⟨w, v⟩∣(w ∈ P ∧ v ∈ P)}
5		19.42v 966	. . . . 5 ⊢ (∃u((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)}) ↔ ((w ∈ P ∧ v ∈ P) ∧ ∃u u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)}))
6		visset 1350	. . . . . . 7 ⊢ w ∈ V
7		visset 1350	. . . . . . 7 ⊢ v ∈ V
8	6, 7	oprvalex 3055	. . . . . 6 ⊢ {x∣∃y ∈ w ∃z ∈ v x = (yGz)} ∈ V
9	8	isseti 1352	. . . . 5 ⊢ ∃u u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)}
10	5, 9	mpbiranr 548	. . . 4 ⊢ (∃u((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)}) ↔ (w ∈ P ∧ v ∈ P))
11	10	biopabi 2103	. . 3 ⊢ {⟨w, v⟩∣∃u((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})} = {⟨w, v⟩∣(w ∈ P ∧ v ∈ P)}
12	4, 11	eqtr4 1122	. 2 ⊢ (P × P) = {⟨w, v⟩∣∃u((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}
13	1, 3, 12	3eqtr4 1126	1 ⊢ dom F = (P × P)

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {copab 2055   × cxp 2408  dom cdm 2410  (class class class)co 3001  {copab2 3002  Pcnp 3779

This theorem is referenced by:  dmplp 3909  dmmp 3910

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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