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Theorem dprdval 19117
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 485 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2 reldmdprd 19111 . . . . . 6 Rel dom DProd
32brrelex2i 5602 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
43adantr 483 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
52brrelex1i 5601 . . . . . 6 (𝐺dom DProd 𝑠𝐺 ∈ V)
6 breq1 5060 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔dom DProd 𝑠𝐺dom DProd 𝑠))
7 oveq1 7155 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8 fveq2 6663 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 dprdval.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
108, 9syl6eqr 2872 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (0g𝑔) = 0 )
1110breq2d 5069 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ( finSupp (0g𝑔) ↔ finSupp 0 ))
1211rabbidv 3479 . . . . . . . . . . 11 (𝑔 = 𝐺 → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} = {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 })
13 oveq1 7155 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
1412, 13mpteq12dv 5142 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
1514rneqd 5801 . . . . . . . . 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 2835 . . . . . . . 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 347 . . . . . . 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 5058 . . . . . . . . 9 (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19 fvex 6676 . . . . . . . . . . . . . . . . 17 (𝑠𝑖) ∈ V
2019rgenw 3148 . . . . . . . . . . . . . . . 16 𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
21 ixpexg 8478 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V)
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
2322mptrabex 6980 . . . . . . . . . . . . . 14 (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2423rnex 7609 . . . . . . . . . . . . 13 ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2524rgen2w 3149 . . . . . . . . . . . 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 19109 . . . . . . . . . . . . 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 7757 . . . . . . . . . . . 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 232 . . . . . . . . . . 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 6517 . . . . . . . . . 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 2902 . . . . . . . . 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 5612 . . . . . . . . 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 299 . . . . . . . 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 7289 . . . . . . . . 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 1443 . . . . . . . 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 219 . . . . . . 7 (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3617, 35vtoclg 3566 . . . . . 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 3785 . . . 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 487 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
4140breq2d 5069 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠𝐺dom DProd 𝑆))
4240oveq2d 7164 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
4340dmeqd 5767 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44 simplr 767 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑆 = 𝐼)
4543, 44eqtrd 2854 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
4645ixpeq1d 8465 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑠𝑖))
4740fveq1d 6665 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠𝑖) = (𝑆𝑖))
4847ixpeq2dv 8469 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖𝐼 (𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
4946, 48eqtrd 2854 . . . . . . . . . 10 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
5049rabeqdv 3483 . . . . . . . . 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, 51syl6eqr 2872 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } = 𝑊)
53 eqidd 2820 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
5452, 53mpteq12dv 5142 . . . . . . 7 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5554rneqd 5801 . . . . . 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 2835 . . . . 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 347 . . . 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 3812 . . 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 234 . 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 398   = wceq 1530  wcel 2107  {cab 2797  wral 3136  {crab 3140  Vcvv 3493  [wsbc 3770  cdif 3931  cin 3933  wss 3934  {csn 4559  cop 4565   cuni 4830   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  dom cdm 5548  ran crn 5549  cima 5551  wf 6344  cfv 6348  (class class class)co 7148  Xcixp 8453   finSupp cfsupp 8825  0gc0g 16705   Σg cgsu 16706  mrClscmrc 16846  Grpcgrp 18095  SubGrpcsubg 18265  Cntzccntz 18437   DProd cdprd 19107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-ixp 8454  df-dprd 19109
This theorem is referenced by:  eldprd  19118  dprdlub  19140
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