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Theorem dprdval 18342
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 473 . 2 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2 reldmdprd 18336 . . . . . 6 Rel dom DProd
32brrelex2i 5129 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
43adantr 481 . . . 4 ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
52brrelexi 5128 . . . . . 6 (𝐺dom DProd 𝑠𝐺 ∈ V)
6 breq1 4626 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔dom DProd 𝑠𝐺dom DProd 𝑠))
7 oveq1 6622 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8 fveq2 6158 . . . . . . . . . . . . . 14 (𝑔 = 𝐺 → (0g𝑔) = (0g𝐺))
9 dprdval.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
108, 9syl6eqr 2673 . . . . . . . . . . . . 13 (𝑔 = 𝐺 → (0g𝑔) = 0 )
1110breq2d 4635 . . . . . . . . . . . 12 (𝑔 = 𝐺 → ( finSupp (0g𝑔) ↔ finSupp 0 ))
1211rabbidv 3181 . . . . . . . . . . 11 (𝑔 = 𝐺 → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} = {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 })
13 oveq1 6622 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
1412, 13mpteq12dv 4703 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
1514rneqd 5323 . . . . . . . . 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 2636 . . . . . . . 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 334 . . . . . . 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 4624 . . . . . . . . 9 (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19 fvex 6168 . . . . . . . . . . . . . . . . 17 (𝑠𝑖) ∈ V
2019rgenw 2920 . . . . . . . . . . . . . . . 16 𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
21 ixpexg 7892 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V)
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15 X𝑖 ∈ dom 𝑠(𝑠𝑖) ∈ V
2322mptrabex 6453 . . . . . . . . . . . . . 14 (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2423rnex 7062 . . . . . . . . . . . . 13 ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
2524rgen2w 2921 . . . . . . . . . . . 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 18334 . . . . . . . . . . . . 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)))
2726fmpt2x 7196 . . . . . . . . . . . 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 220 . . . . . . . . . . 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 6019 . . . . . . . . . 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 2690 . . . . . . . . 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 5141 . . . . . . . . 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 286 . . . . . . . 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 6748 . . . . . . . . 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 1410 . . . . . . . 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 207 . . . . . . 7 (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
3617, 35vtoclg 3256 . . . . . 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 3437 . . . 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 477 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
4140breq2d 4635 . . . . 5 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠𝐺dom DProd 𝑆))
4240oveq2d 6631 . . . . . 6 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
4340dmeqd 5296 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44 simplr 791 . . . . . . . . . . . . 13 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑆 = 𝐼)
4543, 44eqtrd 2655 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
4645ixpeq1d 7880 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑠𝑖))
4740fveq1d 6160 . . . . . . . . . . . 12 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠𝑖) = (𝑆𝑖))
4847ixpeq2dv 7884 . . . . . . . . . . 11 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖𝐼 (𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
4946, 48eqtrd 2655 . . . . . . . . . 10 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠𝑖) = X𝑖𝐼 (𝑆𝑖))
5049rabeqdv 3184 . . . . . . . . 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 2673 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } = 𝑊)
53 eqidd 2622 . . . . . . . 8 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
5452, 53mpteq12dv 4703 . . . . . . 7 (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {X𝑖 ∈ dom 𝑠(𝑠𝑖) ∣ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓𝑊 ↦ (𝐺 Σg 𝑓)))
5554rneqd 5323 . . . . . 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 2636 . . . . 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 334 . . . 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 3459 . . 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 222 . 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 384   = wceq 1480  wcel 1987  {cab 2607  wral 2908  {crab 2912  Vcvv 3190  [wsbc 3422  cdif 3557  cin 3559  wss 3560  {csn 4155  cop 4161   cuni 4409   ciun 4492   class class class wbr 4623  cmpt 4683   × cxp 5082  dom cdm 5084  ran crn 5085  cima 5087  wf 5853  cfv 5857  (class class class)co 6615  Xcixp 7868   finSupp cfsupp 8235  0gc0g 16040   Σg cgsu 16041  mrClscmrc 16183  Grpcgrp 17362  SubGrpcsubg 17528  Cntzccntz 17688   DProd cdprd 18332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-ixp 7869  df-dprd 18334
This theorem is referenced by:  eldprd  18343  dprdlub  18365
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