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Theorem frrlem11 31917
 Description: Lemma for founded recursion. Here, we calculate the value of 𝐹 (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Scott Fenton, 23-Dec-2021.)
Hypotheses
Ref Expression
frrlem10.1 𝑅 Fr 𝐴
frrlem10.2 𝑅 Se 𝐴
frrlem10.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
frrlem10.4 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
frrlem11 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑥,𝐹   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵   𝑓,𝐹
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐹(𝑦)

Proof of Theorem frrlem11
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . 3 𝑦 ∈ V
21eldm2 5354 . 2 (𝑦 ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐹)
3 frrlem10.4 . . . . . . 7 𝐹 = frecs(𝑅, 𝐴, 𝐺)
4 df-frecs 31901 . . . . . . 7 frecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
53, 4eqtri 2673 . . . . . 6 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
65eleq2i 2722 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
7 eluniab 4479 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
86, 7bitri 264 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
9 simpr3 1089 . . . . . . . . . 10 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
10 simpr1 1087 . . . . . . . . . . 11 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑓 Fn 𝑥)
11 simpl 472 . . . . . . . . . . 11 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ⟨𝑦, 𝑧⟩ ∈ 𝑓)
12 fnop 6032 . . . . . . . . . . 11 ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦𝑥)
1310, 11, 12syl2anc 694 . . . . . . . . . 10 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑦𝑥)
14 rsp 2958 . . . . . . . . . 10 (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦𝑥 → (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
159, 13, 14sylc 65 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
16 frrlem10.1 . . . . . . . . . . 11 𝑅 Fr 𝐴
17 frrlem10.2 . . . . . . . . . . 11 𝑅 Se 𝐴
18 frrlem10.3 . . . . . . . . . . 11 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1916, 17, 18, 3frrlem10 31916 . . . . . . . . . 10 Fun 𝐹
20 19.8a 2090 . . . . . . . . . . . . . 14 ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2118abeq2i 2764 . . . . . . . . . . . . . 14 (𝑓𝐵 ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2220, 21sylibr 224 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → 𝑓𝐵)
2322adantl 481 . . . . . . . . . . . 12 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑓𝐵)
24 elssuni 4499 . . . . . . . . . . . 12 (𝑓𝐵𝑓 𝐵)
2523, 24syl 17 . . . . . . . . . . 11 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑓 𝐵)
2618unieqi 4477 . . . . . . . . . . . 12 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
275, 26eqtr4i 2676 . . . . . . . . . . 11 𝐹 = 𝐵
2825, 27syl6sseqr 3685 . . . . . . . . . 10 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑓𝐹)
29 fndm 6028 . . . . . . . . . . . 12 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
3010, 29syl 17 . . . . . . . . . . 11 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → dom 𝑓 = 𝑥)
3113, 30eleqtrrd 2733 . . . . . . . . . 10 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑦 ∈ dom 𝑓)
32 funssfv 6247 . . . . . . . . . 10 ((Fun 𝐹𝑓𝐹𝑦 ∈ dom 𝑓) → (𝐹𝑦) = (𝑓𝑦))
3319, 28, 31, 32mp3an2i 1469 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝑓𝑦))
34 simpr2r 1141 . . . . . . . . . . . . 13 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
35 rsp 2958 . . . . . . . . . . . . 13 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
3634, 13, 35sylc 65 . . . . . . . . . . . 12 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥)
3736, 30sseqtr4d 3675 . . . . . . . . . . 11 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓)
38 fun2ssres 5969 . . . . . . . . . . 11 ((Fun 𝐹𝑓𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
3919, 28, 37, 38mp3an2i 1469 . . . . . . . . . 10 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
4039oveq2d 6706 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
4115, 33, 403eqtr4d 2695 . . . . . . . 8 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ (𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
4241ex 449 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
4342exlimdv 1901 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
4443imp 444 . . . . 5 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
4544exlimiv 1898 . . . 4 (∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
468, 45sylbi 207 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
4746exlimiv 1898 . 2 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
482, 47sylbi 207 1 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941   ⊆ wss 3607  ⟨cop 4216  ∪ cuni 4468   Fr wfr 5099   Se wse 5100  dom cdm 5143   ↾ cres 5145  Predcpred 5717  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690  frecscfrecs 31900 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-trpred 31842  df-frecs 31901 This theorem is referenced by: (None)
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