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Theorem oarec 8513
Description: Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
oarec ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oarec
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7369 . . . 4 (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅))
2 mpteq1 5202 . . . . . . . 8 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)))
3 mpt0 6647 . . . . . . . 8 (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅
42, 3eqtrdi 2789 . . . . . . 7 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
54rneqd 5897 . . . . . 6 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅)
6 rn0 5885 . . . . . 6 ran ∅ = ∅
75, 6eqtrdi 2789 . . . . 5 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
87uneq2d 4127 . . . 4 (𝑧 = ∅ → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅))
91, 8eqeq12d 2749 . . 3 (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅)))
10 oveq2 7369 . . . 4 (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤))
11 mpteq1 5202 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1211rneqd 5897 . . . . 5 (𝑧 = 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1312uneq2d 4127 . . . 4 (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
1410, 13eqeq12d 2749 . . 3 (𝑧 = 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))))
15 oveq2 7369 . . . 4 (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤))
16 mpteq1 5202 . . . . . 6 (𝑧 = suc 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1716rneqd 5897 . . . . 5 (𝑧 = suc 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1817uneq2d 4127 . . . 4 (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
1915, 18eqeq12d 2749 . . 3 (𝑧 = suc 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
20 oveq2 7369 . . . 4 (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵))
21 mpteq1 5202 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2221rneqd 5897 . . . . 5 (𝑧 = 𝐵 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2322uneq2d 4127 . . . 4 (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2420, 23eqeq12d 2749 . . 3 (𝑧 = 𝐵 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
25 oa0 8466 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
26 un0 4354 . . . 4 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtr4di 2791 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪ ∅))
28 uneq1 4120 . . . . . 6 ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}))
29 unass 4130 . . . . . . 7 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
30 rexun 4154 . . . . . . . . . . 11 (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
31 df-suc 6327 . . . . . . . . . . . 12 suc 𝑤 = (𝑤 ∪ {𝑤})
3231rexeqi 3311 . . . . . . . . . . 11 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥))
33 eqid 2733 . . . . . . . . . . . . . 14 (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥))
3433elrnmpt 5915 . . . . . . . . . . . . 13 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥)))
3534elv 3453 . . . . . . . . . . . 12 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥))
36 velsn 4606 . . . . . . . . . . . . 13 (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤))
37 vex 3451 . . . . . . . . . . . . . 14 𝑤 ∈ V
38 oveq2 7369 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
3938eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)))
4037, 39rexsn 4647 . . . . . . . . . . . . 13 (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤))
4136, 40bitr4i 278 . . . . . . . . . . . 12 (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))
4235, 41orbi12i 914 . . . . . . . . . . 11 ((𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
4330, 32, 423bitr4i 303 . . . . . . . . . 10 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
44 eqid 2733 . . . . . . . . . . 11 (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))
45 ovex 7394 . . . . . . . . . . 11 (𝐴 +o 𝑥) ∈ V
4644, 45elrnmpti 5919 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥))
47 elun 4112 . . . . . . . . . 10 (𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
4843, 46, 473bitr4i 303 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
4948eqriv 2730 . . . . . . . 8 ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})
5049uneq2i 4124 . . . . . . 7 (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
5129, 50eqtr4i 2764 . . . . . 6 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
5228, 51eqtrdi 2789 . . . . 5 ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
53 oasuc 8474 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤))
54 df-suc 6327 . . . . . . 7 suc (𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})
5553, 54eqtrdi 2789 . . . . . 6 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}))
5655eqeq1d 2735 . . . . 5 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) ↔ ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
5752, 56imbitrrid 245 . . . 4 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
5857expcom 415 . . 3 (𝑤 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))))
59 vex 3451 . . . . . . . 8 𝑧 ∈ V
60 oalim 8482 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6159, 60mpanr1 702 . . . . . . 7 ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6261ancoms 460 . . . . . 6 ((Lim 𝑧𝐴 ∈ On) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6362adantr 482 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
64 iuneq2 4977 . . . . . 6 (∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → 𝑤𝑧 (𝐴 +o 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
6564adantl 483 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → 𝑤𝑧 (𝐴 +o 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
66 iunun 5057 . . . . . . 7 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
67 0ellim 6384 . . . . . . . . 9 (Lim 𝑧 → ∅ ∈ 𝑧)
68 ne0i 4298 . . . . . . . . 9 (∅ ∈ 𝑧𝑧 ≠ ∅)
69 iunconst 4967 . . . . . . . . 9 (𝑧 ≠ ∅ → 𝑤𝑧 𝐴 = 𝐴)
7067, 68, 693syl 18 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 𝐴 = 𝐴)
71 df-rex 3071 . . . . . . . . . . . . . 14 (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7235, 71bitri 275 . . . . . . . . . . . . 13 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7372rexbii 3094 . . . . . . . . . . . 12 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
74 eluni2 4873 . . . . . . . . . . . . . . . 16 (𝑥 𝑧 ↔ ∃𝑤𝑧 𝑥𝑤)
7574anbi1i 625 . . . . . . . . . . . . . . 15 ((𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
76 r19.41v 3182 . . . . . . . . . . . . . . 15 (∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7775, 76bitr4i 278 . . . . . . . . . . . . . 14 ((𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7877exbii 1851 . . . . . . . . . . . . 13 (∃𝑥(𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
79 df-rex 3071 . . . . . . . . . . . . 13 (∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 𝑧𝑦 = (𝐴 +o 𝑥)))
80 rexcom4 3270 . . . . . . . . . . . . 13 (∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
8178, 79, 803bitr4i 303 . . . . . . . . . . . 12 (∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
8273, 81bitr4i 278 . . . . . . . . . . 11 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥))
83 limuni 6382 . . . . . . . . . . . 12 (Lim 𝑧𝑧 = 𝑧)
8483rexeqdv 3313 . . . . . . . . . . 11 (Lim 𝑧 → (∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥)))
8582, 84bitr4id 290 . . . . . . . . . 10 (Lim 𝑧 → (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥)))
86 eliun 4962 . . . . . . . . . 10 (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
87 eqid 2733 . . . . . . . . . . 11 (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑧 ↦ (𝐴 +o 𝑥))
8887, 45elrnmpti 5919 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥))
8985, 86, 883bitr4g 314 . . . . . . . . 9 (Lim 𝑧 → (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9089eqrdv 2731 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)))
9170, 90uneq12d 4128 . . . . . . 7 (Lim 𝑧 → ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9266, 91eqtrid 2785 . . . . . 6 (Lim 𝑧 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9392ad2antrr 725 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9463, 65, 933eqtrd 2777 . . . 4 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9594exp31 421 . . 3 (Lim 𝑧 → (𝐴 ∈ On → (∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))))
969, 14, 19, 24, 27, 58, 95tfinds3 7805 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
9796impcom 409 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wne 2940  wral 3061  wrex 3070  Vcvv 3447  cun 3912  c0 4286  {csn 4590   cuni 4869   ciun 4958  cmpt 5192  ran crn 5638  Oncon0 6321  Lim wlim 6322  suc csuc 6323  (class class class)co 7361   +o coa 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420
This theorem is referenced by:  oacomf1o  8516  onadju  10137
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