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Theorem dprdval 20024
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 482 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2 reldmdprd 20018 . . . . . 6 Rel dom DProd
32brrelex2i 5741 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
43adantr 480 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
52brrelex1i 5740 . . . . . 6 (𝐺dom DProd 𝑠𝐺 ∈ V)
6 breq1 5145 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔dom DProd 𝑠𝐺dom DProd 𝑠))
7 oveq1 7439 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8 fveq2 6905 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 dprdval.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
108, 9eqtr4di 2794 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (0g𝑔) = 0 )
1110breq2d 5154 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ( finSupp (0g𝑔) ↔ finSupp 0 ))
1211rabbidv 3443 . . . . . . . . . . 11 (𝑔 = 𝐺 → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} = {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 })
13 oveq1 7439 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
1412, 13mpteq12dv 5232 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
1514rneqd 5948 . . . . . . . . 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 2752 . . . . . . . 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 344 . . . . . . 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 5143 . . . . . . . . 9 (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19 fvex 6918 . . . . . . . . . . . . . . . . 17 (𝑠𝑖) ∈ V
2019rgenw 3064 . . . . . . . . . . . . . . . 16 𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
21 ixpexg 8963 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V)
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
2322mptrabex 7246 . . . . . . . . . . . . . 14 (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2423rnex 7933 . . . . . . . . . . . . 13 ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2524rgen2w 3065 . . . . . . . . . . . 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 20016 . . . . . . . . . . . . 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 8093 . . . . . . . . . . . 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 230 . . . . . . . . . . 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 6746 . . . . . . . . . 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 2832 . . . . . . . . 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 5751 . . . . . . . . 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 7581 . . . . . . . . 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 217 . . . . . . 7 (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3617, 35vtoclg 3553 . . . . . 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 3802 . . . 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 484 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
4140breq2d 5154 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠𝐺dom DProd 𝑆))
4240oveq2d 7448 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
4340dmeqd 5915 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
4645ixpeq1d 8950 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑠𝑖))
4740fveq1d 6907 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠𝑖) = (𝑆𝑖))
4847ixpeq2dv 8954 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖𝐼 (𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
4946, 48eqtrd 2776 . . . . . . . . . 10 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
5049rabeqdv 3451 . . . . . . . . 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 2794 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } = 𝑊)
53 eqidd 2737 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
5452, 53mpteq12dv 5232 . . . . . . 7 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5554rneqd 5948 . . . . . 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 2752 . . . . 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 344 . . . 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 3831 . . 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 232 . 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 395   = wceq 1539  wcel 2107  {cab 2713  wral 3060  {crab 3435  Vcvv 3479  [wsbc 3787  cdif 3947  cin 3949  wss 3950  {csn 4625  cop 4631   cuni 4906   ciun 4990   class class class wbr 5142  cmpt 5224   × cxp 5682  dom cdm 5684  ran crn 5685  cima 5687  wf 6556  cfv 6560  (class class class)co 7432  Xcixp 8938   finSupp cfsupp 9402  0gc0g 17485   Σg cgsu 17486  mrClscmrc 17627  Grpcgrp 18952  SubGrpcsubg 19139  Cntzccntz 19334   DProd cdprd 20014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-ixp 8939  df-dprd 20016
This theorem is referenced by:  eldprd  20025  dprdlub  20047
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