MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oarec Structured version   Visualization version   GIF version

Theorem oarec 8600
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 7439 . . . 4 (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅))
2 mpteq1 5235 . . . . . . . 8 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)))
3 mpt0 6710 . . . . . . . 8 (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅
42, 3eqtrdi 2793 . . . . . . 7 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
54rneqd 5949 . . . . . 6 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅)
6 rn0 5936 . . . . . 6 ran ∅ = ∅
75, 6eqtrdi 2793 . . . . 5 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
87uneq2d 4168 . . . 4 (𝑧 = ∅ → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅))
91, 8eqeq12d 2753 . . 3 (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅)))
10 oveq2 7439 . . . 4 (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤))
11 mpteq1 5235 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1211rneqd 5949 . . . . 5 (𝑧 = 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1312uneq2d 4168 . . . 4 (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
1410, 13eqeq12d 2753 . . 3 (𝑧 = 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))))
15 oveq2 7439 . . . 4 (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤))
16 mpteq1 5235 . . . . . 6 (𝑧 = suc 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1716rneqd 5949 . . . . 5 (𝑧 = suc 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1817uneq2d 4168 . . . 4 (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
1915, 18eqeq12d 2753 . . 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 7439 . . . 4 (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵))
21 mpteq1 5235 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2221rneqd 5949 . . . . 5 (𝑧 = 𝐵 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2322uneq2d 4168 . . . 4 (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2420, 23eqeq12d 2753 . . 3 (𝑧 = 𝐵 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
25 oa0 8554 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
26 un0 4394 . . . 4 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtr4di 2795 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪ ∅))
28 uneq1 4161 . . . . . 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 4172 . . . . . . 7 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
30 rexun 4196 . . . . . . . . . . 11 (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
31 df-suc 6390 . . . . . . . . . . . 12 suc 𝑤 = (𝑤 ∪ {𝑤})
3231rexeqi 3325 . . . . . . . . . . 11 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥))
33 eqid 2737 . . . . . . . . . . . . . 14 (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥))
3433elrnmpt 5969 . . . . . . . . . . . . 13 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥)))
3534elv 3485 . . . . . . . . . . . 12 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥))
36 velsn 4642 . . . . . . . . . . . . 13 (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤))
37 vex 3484 . . . . . . . . . . . . . 14 𝑤 ∈ V
38 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
3938eqeq2d 2748 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)))
4037, 39rexsn 4682 . . . . . . . . . . . . 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 915 . . . . . . . . . . 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 2737 . . . . . . . . . . 11 (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))
45 ovex 7464 . . . . . . . . . . 11 (𝐴 +o 𝑥) ∈ V
4644, 45elrnmpti 5973 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥))
47 elun 4153 . . . . . . . . . 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 2734 . . . . . . . 8 ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})
5049uneq2i 4165 . . . . . . 7 (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
5129, 50eqtr4i 2768 . . . . . 6 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
5228, 51eqtrdi 2793 . . . . 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 8562 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤))
54 df-suc 6390 . . . . . . 7 suc (𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})
5553, 54eqtrdi 2793 . . . . . 6 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}))
5655eqeq1d 2739 . . . . 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 246 . . . 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 413 . . 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 3484 . . . . . . . 8 𝑧 ∈ V
60 oalim 8570 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6159, 60mpanr1 703 . . . . . . 7 ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6261ancoms 458 . . . . . 6 ((Lim 𝑧𝐴 ∈ On) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6362adantr 480 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
64 iuneq2 5011 . . . . . 6 (∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → 𝑤𝑧 (𝐴 +o 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
6564adantl 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))))
66 iunun 5093 . . . . . . 7 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
67 0ellim 6447 . . . . . . . . 9 (Lim 𝑧 → ∅ ∈ 𝑧)
68 ne0i 4341 . . . . . . . . 9 (∅ ∈ 𝑧𝑧 ≠ ∅)
69 iunconst 5001 . . . . . . . . 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 4911 . . . . . . . . . . . . . . . 16 (𝑥 𝑧 ↔ ∃𝑤𝑧 𝑥𝑤)
7574anbi1i 624 . . . . . . . . . . . . . . 15 ((𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
76 r19.41v 3189 . . . . . . . . . . . . . . 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 1848 . . . . . . . . . . . . 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 3288 . . . . . . . . . . . . 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 6445 . . . . . . . . . . . 12 (Lim 𝑧𝑧 = 𝑧)
8483rexeqdv 3327 . . . . . . . . . . 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 4995 . . . . . . . . . 10 (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
87 eqid 2737 . . . . . . . . . . 11 (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑧 ↦ (𝐴 +o 𝑥))
8887, 45elrnmpti 5973 . . . . . . . . . 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 2735 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)))
9170, 90uneq12d 4169 . . . . . . 7 (Lim 𝑧 → ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9266, 91eqtrid 2789 . . . . . 6 (Lim 𝑧 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9392ad2antrr 726 . . . . 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 2781 . . . 4 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9594exp31 419 . . 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 7886 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
9796impcom 407 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 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cun 3949  c0 4333  {csn 4626   cuni 4907   ciun 4991  cmpt 5225  ran crn 5686  Oncon0 6384  Lim wlim 6385  suc csuc 6386  (class class class)co 7431   +o coa 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  oacomf1o  8603  onadju  10234
  Copyright terms: Public domain W3C validator