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Theorem dprdval 19244
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 486 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2 reldmdprd 19238 . . . . . 6 Rel dom DProd
32brrelex2i 5580 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
43adantr 484 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
52brrelex1i 5579 . . . . . 6 (𝐺dom DProd 𝑠𝐺 ∈ V)
6 breq1 5033 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔dom DProd 𝑠𝐺dom DProd 𝑠))
7 oveq1 7177 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8 fveq2 6674 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 dprdval.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
108, 9eqtr4di 2791 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (0g𝑔) = 0 )
1110breq2d 5042 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ( finSupp (0g𝑔) ↔ finSupp 0 ))
1211rabbidv 3381 . . . . . . . . . . 11 (𝑔 = 𝐺 → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} = {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 })
13 oveq1 7177 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
1412, 13mpteq12dv 5115 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
1514rneqd 5781 . . . . . . . . 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 2754 . . . . . . . 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 348 . . . . . . 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 5031 . . . . . . . . 9 (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19 fvex 6687 . . . . . . . . . . . . . . . . 17 (𝑠𝑖) ∈ V
2019rgenw 3065 . . . . . . . . . . . . . . . 16 𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
21 ixpexg 8532 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V)
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
2322mptrabex 6998 . . . . . . . . . . . . . 14 (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2423rnex 7643 . . . . . . . . . . . . 13 ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2524rgen2w 3066 . . . . . . . . . . . 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 19236 . . . . . . . . . . . . 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 7790 . . . . . . . . . . . 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 233 . . . . . . . . . . 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 6516 . . . . . . . . . 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 2824 . . . . . . . . 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 5590 . . . . . . . . 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 300 . . . . . . . 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 7312 . . . . . . . . 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 220 . . . . . . 7 (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3617, 35vtoclg 3470 . . . . . 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 3695 . . . 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 488 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
4140breq2d 5042 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠𝐺dom DProd 𝑆))
4240oveq2d 7186 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
4340dmeqd 5748 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44 simplr 769 . . . . . . . . . . . . 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 8519 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑠𝑖))
4740fveq1d 6676 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠𝑖) = (𝑆𝑖))
4847ixpeq2dv 8523 . . . . . . . . . . 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 3386 . . . . . . . . 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 2739 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
5452, 53mpteq12dv 5115 . . . . . . 7 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5554rneqd 5781 . . . . . 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 2754 . . . . 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 348 . . . 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 3724 . . 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 235 . 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 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053  {crab 3057  Vcvv 3398  [wsbc 3680  cdif 3840  cin 3842  wss 3843  {csn 4516  cop 4522   cuni 4796   ciun 4881   class class class wbr 5030  cmpt 5110   × cxp 5523  dom cdm 5525  ran crn 5526  cima 5528  wf 6335  cfv 6339  (class class class)co 7170  Xcixp 8507   finSupp cfsupp 8906  0gc0g 16816   Σg cgsu 16817  mrClscmrc 16957  Grpcgrp 18219  SubGrpcsubg 18391  Cntzccntz 18563   DProd cdprd 19234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-ixp 8508  df-dprd 19236
This theorem is referenced by:  eldprd  19245  dprdlub  19267
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