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Theorem oarec 7529
 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 6557 . . . 4 (𝑧 = ∅ → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 ∅))
2 mpteq1 4665 . . . . . . . 8 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)))
3 mpt0 5934 . . . . . . . 8 (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)) = ∅
42, 3syl6eq 2660 . . . . . . 7 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
54rneqd 5274 . . . . . 6 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran ∅)
6 rn0 5298 . . . . . 6 ran ∅ = ∅
75, 6syl6eq 2660 . . . . 5 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
87uneq2d 3729 . . . 4 (𝑧 = ∅ → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ∅))
91, 8eqeq12d 2625 . . 3 (𝑧 = ∅ → ((𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) ↔ (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅)))
10 oveq2 6557 . . . 4 (𝑧 = 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤))
11 mpteq1 4665 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))
1211rneqd 5274 . . . . 5 (𝑧 = 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))
1312uneq2d 3729 . . . 4 (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))))
1410, 13eqeq12d 2625 . . 3 (𝑧 = 𝑤 → ((𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) ↔ (𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))))
15 oveq2 6557 . . . 4 (𝑧 = suc 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 suc 𝑤))
16 mpteq1 4665 . . . . . 6 (𝑧 = suc 𝑤 → (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
1716rneqd 5274 . . . . 5 (𝑧 = suc 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
1817uneq2d 3729 . . . 4 (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))))
1915, 18eqeq12d 2625 . . 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 6557 . . . 4 (𝑧 = 𝐵 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝐵))
21 mpteq1 4665 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))
2221rneqd 5274 . . . . 5 (𝑧 = 𝐵 → ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))
2322uneq2d 3729 . . . 4 (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))))
2420, 23eqeq12d 2625 . . 3 (𝑧 = 𝐵 → ((𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))) ↔ (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))))
25 oa0 7483 . . . 4 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
26 un0 3919 . . . 4 (𝐴 ∪ ∅) = 𝐴
2725, 26syl6eqr 2662 . . 3 (𝐴 ∈ On → (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅))
28 uneq1 3722 . . . . . 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 3732 . . . . . . 7 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
30 rexun 3755 . . . . . . . . . . 11 (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +𝑜 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥)))
31 df-suc 5646 . . . . . . . . . . . 12 suc 𝑤 = (𝑤 ∪ {𝑤})
3231rexeqi 3120 . . . . . . . . . . 11 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥))
33 vex 3176 . . . . . . . . . . . . 13 𝑦 ∈ V
34 eqid 2610 . . . . . . . . . . . . . 14 (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))
3534elrnmpt 5293 . . . . . . . . . . . . 13 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +𝑜 𝑥)))
3633, 35ax-mp 5 . . . . . . . . . . . 12 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +𝑜 𝑥))
37 velsn 4141 . . . . . . . . . . . . 13 (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ 𝑦 = (𝐴 +𝑜 𝑤))
38 vex 3176 . . . . . . . . . . . . . 14 𝑤 ∈ V
39 oveq2 6557 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
4039eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤)))
4138, 40rexsn 4170 . . . . . . . . . . . . 13 (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤))
4237, 41bitr4i 266 . . . . . . . . . . . 12 (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥))
4336, 42orbi12i 542 . . . . . . . . . . 11 ((𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +𝑜 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥)))
4430, 32, 433bitr4i 291 . . . . . . . . . 10 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}))
45 eqid 2610 . . . . . . . . . . 11 (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))
46 ovex 6577 . . . . . . . . . . 11 (𝐴 +𝑜 𝑥) ∈ V
4745, 46elrnmpti 5297 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥))
48 elun 3715 . . . . . . . . . 10 (𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}))
4944, 47, 483bitr4i 291 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
5049eqriv 2607 . . . . . . . 8 ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)})
5150uneq2i 3726 . . . . . . 7 (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
5229, 51eqtr4i 2635 . . . . . 6 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
5328, 52syl6eq 2660 . . . . 5 ((𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) → ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))))
54 oasuc 7491 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = suc (𝐴 +𝑜 𝑤))
55 df-suc 5646 . . . . . . 7 suc (𝐴 +𝑜 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)})
5654, 55syl6eq 2660 . . . . . 6 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)}))
5756eqeq1d 2612 . . . . 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)))))
5853, 57syl5ibr 235 . . . 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)))))
5958expcom 450 . . 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))))))
60 vex 3176 . . . . . . . 8 𝑧 ∈ V
61 oalim 7499 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +𝑜 𝑧) = 𝑤𝑧 (𝐴 +𝑜 𝑤))
6260, 61mpanr1 715 . . . . . . 7 ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +𝑜 𝑧) = 𝑤𝑧 (𝐴 +𝑜 𝑤))
6362ancoms 468 . . . . . 6 ((Lim 𝑧𝐴 ∈ On) → (𝐴 +𝑜 𝑧) = 𝑤𝑧 (𝐴 +𝑜 𝑤))
6463adantr 480 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))) → (𝐴 +𝑜 𝑧) = 𝑤𝑧 (𝐴 +𝑜 𝑤))
65 iuneq2 4473 . . . . . 6 (∀𝑤𝑧 (𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) → 𝑤𝑧 (𝐴 +𝑜 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))))
6665adantl 481 . . . . 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))))
67 iunun 4540 . . . . . . 7 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) = ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))
68 0ellim 5704 . . . . . . . . 9 (Lim 𝑧 → ∅ ∈ 𝑧)
69 ne0i 3880 . . . . . . . . 9 (∅ ∈ 𝑧𝑧 ≠ ∅)
70 iunconst 4465 . . . . . . . . 9 (𝑧 ≠ ∅ → 𝑤𝑧 𝐴 = 𝐴)
7168, 69, 703syl 18 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 𝐴 = 𝐴)
72 limuni 5702 . . . . . . . . . . . 12 (Lim 𝑧𝑧 = 𝑧)
7372rexeqdv 3122 . . . . . . . . . . 11 (Lim 𝑧 → (∃𝑥𝑧 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥)))
74 df-rex 2902 . . . . . . . . . . . . . 14 (∃𝑥𝑤 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
7536, 74bitri 263 . . . . . . . . . . . . 13 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
7675rexbii 3023 . . . . . . . . . . . 12 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
77 eluni2 4376 . . . . . . . . . . . . . . . 16 (𝑥 𝑧 ↔ ∃𝑤𝑧 𝑥𝑤)
7877anbi1i 727 . . . . . . . . . . . . . . 15 ((𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
79 r19.41v 3070 . . . . . . . . . . . . . . 15 (∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
8078, 79bitr4i 266 . . . . . . . . . . . . . 14 ((𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
8180exbii 1764 . . . . . . . . . . . . 13 (∃𝑥(𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
82 df-rex 2902 . . . . . . . . . . . . 13 (∃𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥)))
83 rexcom4 3198 . . . . . . . . . . . . 13 (∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
8481, 82, 833bitr4i 291 . . . . . . . . . . . 12 (∃𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +𝑜 𝑥)))
8576, 84bitr4i 266 . . . . . . . . . . 11 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +𝑜 𝑥))
8673, 85syl6rbbr 278 . . . . . . . . . 10 (Lim 𝑧 → (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +𝑜 𝑥)))
87 eliun 4460 . . . . . . . . . 10 (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))
88 eqid 2610 . . . . . . . . . . 11 (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))
8988, 46elrnmpti 5297 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +𝑜 𝑥))
9086, 87, 893bitr4g 302 . . . . . . . . 9 (Lim 𝑧 → (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))))
9190eqrdv 2608 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥)))
9271, 91uneq12d 3730 . . . . . . 7 (Lim 𝑧 → ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))))
9367, 92syl5eq 2656 . . . . . 6 (Lim 𝑧 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))))
9493ad2antrr 758 . . . . 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))))
9564, 66, 943eqtrd 2648 . . . 4 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥)))) → (𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))))
9695exp31 628 . . 3 (Lim 𝑧 → (𝐴 ∈ On → (∀𝑤𝑧 (𝐴 +𝑜 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +𝑜 𝑥))) → (𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +𝑜 𝑥))))))
979, 14, 19, 24, 27, 59, 96tfinds3 6956 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))))
9897impcom 445 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ran crn 5039  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451 This theorem is referenced by:  oacomf1o  7532  onacda  8902
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