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Theorem dprdval 19868
Description: The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdval.0 0 = (0g𝐺)
dprdval.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
Assertion
Ref Expression
dprdval ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
Distinct variable groups:   𝑓,,𝑖,𝐼   𝑆,𝑓,,𝑖   𝑓,𝐺,,𝑖
Allowed substitution hints:   𝑊(𝑓,,𝑖)   0 (𝑓,,𝑖)

Proof of Theorem dprdval
Dummy variables 𝑔 𝑠 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 484 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2 reldmdprd 19862 . . . . . 6 Rel dom DProd
32brrelex2i 5732 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
43adantr 482 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
52brrelex1i 5731 . . . . . 6 (𝐺dom DProd 𝑠𝐺 ∈ V)
6 breq1 5151 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔dom DProd 𝑠𝐺dom DProd 𝑠))
7 oveq1 7413 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8 fveq2 6889 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 dprdval.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
108, 9eqtr4di 2791 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (0g𝑔) = 0 )
1110breq2d 5160 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ( finSupp (0g𝑔) ↔ finSupp 0 ))
1211rabbidv 3441 . . . . . . . . . . 11 (𝑔 = 𝐺 → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} = {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 })
13 oveq1 7413 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
1412, 13mpteq12dv 5239 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
1514rneqd 5936 . . . . . . . . 9 (𝑔 = 𝐺 → ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
167, 15eqeq12d 2749 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ↔ (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
176, 16imbi12d 345 . . . . . . 7 (𝑔 = 𝐺 → ((𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓))) ↔ (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))))
18 df-br 5149 . . . . . . . . 9 (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19 fvex 6902 . . . . . . . . . . . . . . . . 17 (𝑠𝑖) ∈ V
2019rgenw 3066 . . . . . . . . . . . . . . . 16 𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
21 ixpexg 8913 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V)
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
2322mptrabex 7224 . . . . . . . . . . . . . 14 (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2423rnex 7900 . . . . . . . . . . . . 13 ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2524rgen2w 3067 . . . . . . . . . . . 12 𝑔 ∈ Grp ∀𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
26 df-dprd 19860 . . . . . . . . . . . . 13 DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
2726fmpox 8050 . . . . . . . . . . . 12 (∀𝑔 ∈ Grp ∀𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V ↔ DProd : 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))})⟶V)
2825, 27mpbi 229 . . . . . . . . . . 11 DProd : 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))})⟶V
2928fdmi 6727 . . . . . . . . . 10 dom DProd = 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))})
3029eleq2i 2826 . . . . . . . . 9 (⟨𝑔, 𝑠⟩ ∈ dom DProd ↔ ⟨𝑔, 𝑠⟩ ∈ 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}))
31 opeliunxp 5742 . . . . . . . . 9 (⟨𝑔, 𝑠⟩ ∈ 𝑔 ∈ Grp ({𝑔} × { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}) ↔ (𝑔 ∈ Grp ∧ 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}))
3218, 30, 313bitri 297 . . . . . . . 8 (𝑔dom DProd 𝑠 ↔ (𝑔 ∈ Grp ∧ 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}))
3326ovmpt4g 7552 . . . . . . . . 9 ((𝑔 ∈ Grp ∧ 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))} ∧ ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V) → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3424, 33mp3an3 1451 . . . . . . . 8 ((𝑔 ∈ Grp ∧ 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑖})(𝑖) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑖})))) = {(0g𝑔)}))}) → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3532, 34sylbi 216 . . . . . . 7 (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3617, 35vtoclg 3557 . . . . . 6 (𝐺 ∈ V → (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
375, 36mpcom 38 . . . . 5 (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
3837sbcth 3792 . . . 4 (𝑆 ∈ V → [𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
394, 38syl 17 . . 3 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → [𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
40 simpr 486 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
4140breq2d 5160 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠𝐺dom DProd 𝑆))
4240oveq2d 7422 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
4340dmeqd 5904 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44 simplr 768 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑆 = 𝐼)
4543, 44eqtrd 2773 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
4645ixpeq1d 8900 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑠𝑖))
4740fveq1d 6891 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠𝑖) = (𝑆𝑖))
4847ixpeq2dv 8904 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖𝐼 (𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
4946, 48eqtrd 2773 . . . . . . . . . 10 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
5049rabeqdv 3448 . . . . . . . . 9 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 })
51 dprdval.w . . . . . . . . 9 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
5250, 51eqtr4di 2791 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } = 𝑊)
53 eqidd 2734 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
5452, 53mpteq12dv 5239 . . . . . . 7 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5554rneqd 5936 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5642, 55eqeq12d 2749 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ((𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) ↔ (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
5741, 56imbi12d 345 . . . 4 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ((𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))) ↔ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))))
584, 57sbcied 3822 . . 3 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → ([𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓))) ↔ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))))
5939, 58mpbid 231 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓))))
601, 59mpd 15 1 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3062  {crab 3433  Vcvv 3475  [wsbc 3777  cdif 3945  cin 3947  wss 3948  {csn 4628  cop 4634   cuni 4908   ciun 4997   class class class wbr 5148  cmpt 5231   × cxp 5674  dom cdm 5676  ran crn 5677  cima 5679  wf 6537  cfv 6541  (class class class)co 7406  Xcixp 8888   finSupp cfsupp 9358  0gc0g 17382   Σg cgsu 17383  mrClscmrc 17524  Grpcgrp 18816  SubGrpcsubg 18995  Cntzccntz 19174   DProd cdprd 19858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-ixp 8889  df-dprd 19860
This theorem is referenced by:  eldprd  19869  dprdlub  19891
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