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Theorem wuncval2 10688
Description: Our earlier expression for a containing weak universe is in fact the weak universe closure. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wuncval2.f 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰)
wuncval2.u π‘ˆ = βˆͺ ran 𝐹
Assertion
Ref Expression
wuncval2 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) = π‘ˆ)
Distinct variable groups:   π‘₯,𝑦,𝑧   π‘₯,𝐴,𝑦   π‘₯,𝑉,𝑦
Allowed substitution hints:   𝐴(𝑧)   π‘ˆ(π‘₯,𝑦,𝑧)   𝐹(π‘₯,𝑦,𝑧)   𝑉(𝑧)

Proof of Theorem wuncval2
Dummy variables 𝑣 𝑒 𝑀 π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wuncval2.f . . . 4 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰)
2 wuncval2.u . . . 4 π‘ˆ = βˆͺ ran 𝐹
31, 2wunex2 10679 . . 3 (𝐴 ∈ 𝑉 β†’ (π‘ˆ ∈ WUni ∧ 𝐴 βŠ† π‘ˆ))
4 wuncss 10686 . . 3 ((π‘ˆ ∈ WUni ∧ 𝐴 βŠ† π‘ˆ) β†’ (wUniClβ€˜π΄) βŠ† π‘ˆ)
53, 4syl 17 . 2 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) βŠ† π‘ˆ)
6 frfnom 8382 . . . . . 6 (rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰) Fn Ο‰
71fneq1i 6600 . . . . . 6 (𝐹 Fn Ο‰ ↔ (rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰) Fn Ο‰)
86, 7mpbir 230 . . . . 5 𝐹 Fn Ο‰
9 fniunfv 7195 . . . . 5 (𝐹 Fn Ο‰ β†’ βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š) = βˆͺ ran 𝐹)
108, 9ax-mp 5 . . . 4 βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š) = βˆͺ ran 𝐹
112, 10eqtr4i 2764 . . 3 π‘ˆ = βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š)
12 fveq2 6843 . . . . . . . 8 (π‘š = βˆ… β†’ (πΉβ€˜π‘š) = (πΉβ€˜βˆ…))
1312sseq1d 3976 . . . . . . 7 (π‘š = βˆ… β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄)))
14 fveq2 6843 . . . . . . . 8 (π‘š = 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜π‘›))
1514sseq1d 3976 . . . . . . 7 (π‘š = 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)))
16 fveq2 6843 . . . . . . . 8 (π‘š = suc 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜suc 𝑛))
1716sseq1d 3976 . . . . . . 7 (π‘š = suc 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
18 1on 8425 . . . . . . . . . 10 1o ∈ On
19 unexg 7684 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) β†’ (𝐴 βˆͺ 1o) ∈ V)
2018, 19mpan2 690 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) ∈ V)
211fveq1i 6844 . . . . . . . . . 10 (πΉβ€˜βˆ…) = ((rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰)β€˜βˆ…)
22 fr0g 8383 . . . . . . . . . 10 ((𝐴 βˆͺ 1o) ∈ V β†’ ((rec((𝑧 ∈ V ↦ ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))), (𝐴 βˆͺ 1o)) β†Ύ Ο‰)β€˜βˆ…) = (𝐴 βˆͺ 1o))
2321, 22eqtrid 2785 . . . . . . . . 9 ((𝐴 βˆͺ 1o) ∈ V β†’ (πΉβ€˜βˆ…) = (𝐴 βˆͺ 1o))
2420, 23syl 17 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ (πΉβ€˜βˆ…) = (𝐴 βˆͺ 1o))
25 wuncid 10684 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 𝐴 βŠ† (wUniClβ€˜π΄))
26 df1o2 8420 . . . . . . . . . 10 1o = {βˆ…}
27 wunccl 10685 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) ∈ WUni)
2827wun0 10659 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ βˆ… ∈ (wUniClβ€˜π΄))
2928snssd 4770 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ {βˆ…} βŠ† (wUniClβ€˜π΄))
3026, 29eqsstrid 3993 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 1o βŠ† (wUniClβ€˜π΄))
3125, 30unssd 4147 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) βŠ† (wUniClβ€˜π΄))
3224, 31eqsstrd 3983 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄))
33 simplr 768 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ 𝑛 ∈ Ο‰)
34 fvex 6856 . . . . . . . . . . . . 13 (πΉβ€˜π‘›) ∈ V
3534uniex 7679 . . . . . . . . . . . . 13 βˆͺ (πΉβ€˜π‘›) ∈ V
3634, 35unex 7681 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) ∈ V
37 prex 5390 . . . . . . . . . . . . . 14 {𝒫 𝑒, βˆͺ 𝑒} ∈ V
3834mptex 7174 . . . . . . . . . . . . . . 15 (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
3938rnex 7850 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
4037, 39unex 7681 . . . . . . . . . . . . 13 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4134, 40iunex 7902 . . . . . . . . . . . 12 βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4236, 41unex 7681 . . . . . . . . . . 11 (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))) ∈ V
43 id 22 . . . . . . . . . . . . . 14 (𝑀 = 𝑧 β†’ 𝑀 = 𝑧)
44 unieq 4877 . . . . . . . . . . . . . 14 (𝑀 = 𝑧 β†’ βˆͺ 𝑀 = βˆͺ 𝑧)
4543, 44uneq12d 4125 . . . . . . . . . . . . 13 (𝑀 = 𝑧 β†’ (𝑀 βˆͺ βˆͺ 𝑀) = (𝑧 βˆͺ βˆͺ 𝑧))
46 pweq 4575 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ 𝒫 𝑒 = 𝒫 π‘₯)
47 unieq 4877 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ βˆͺ 𝑒 = βˆͺ π‘₯)
4846, 47preq12d 4703 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ {𝒫 𝑒, βˆͺ 𝑒} = {𝒫 π‘₯, βˆͺ π‘₯})
49 preq1 4695 . . . . . . . . . . . . . . . . . 18 (𝑒 = π‘₯ β†’ {𝑒, 𝑣} = {π‘₯, 𝑣})
5049mpteq2dv 5208 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5150rneqd 5894 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5248, 51uneq12d 4125 . . . . . . . . . . . . . . 15 (𝑒 = π‘₯ β†’ ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣})) = ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣})))
5352cbviunv 5001 . . . . . . . . . . . . . 14 βˆͺ 𝑒 ∈ 𝑀 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣})) = βˆͺ π‘₯ ∈ 𝑀 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
54 preq2 4696 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 β†’ {π‘₯, 𝑣} = {π‘₯, 𝑦})
5554cbvmptv 5219 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦})
56 mpteq1 5199 . . . . . . . . . . . . . . . . . 18 (𝑀 = 𝑧 β†’ (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5755, 56eqtrid 2785 . . . . . . . . . . . . . . . . 17 (𝑀 = 𝑧 β†’ (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5857rneqd 5894 . . . . . . . . . . . . . . . 16 (𝑀 = 𝑧 β†’ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5958uneq2d 4124 . . . . . . . . . . . . . . 15 (𝑀 = 𝑧 β†’ ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣})) = ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))
6043, 59iuneq12d 4983 . . . . . . . . . . . . . 14 (𝑀 = 𝑧 β†’ βˆͺ π‘₯ ∈ 𝑀 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣})) = βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))
6153, 60eqtrid 2785 . . . . . . . . . . . . 13 (𝑀 = 𝑧 β†’ βˆͺ 𝑒 ∈ 𝑀 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣})) = βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦})))
6245, 61uneq12d 4125 . . . . . . . . . . . 12 (𝑀 = 𝑧 β†’ ((𝑀 βˆͺ βˆͺ 𝑀) βˆͺ βˆͺ 𝑒 ∈ 𝑀 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}))) = ((𝑧 βˆͺ βˆͺ 𝑧) βˆͺ βˆͺ π‘₯ ∈ 𝑧 ({𝒫 π‘₯, βˆͺ π‘₯} βˆͺ ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))))
63 id 22 . . . . . . . . . . . . . 14 (𝑀 = (πΉβ€˜π‘›) β†’ 𝑀 = (πΉβ€˜π‘›))
64 unieq 4877 . . . . . . . . . . . . . 14 (𝑀 = (πΉβ€˜π‘›) β†’ βˆͺ 𝑀 = βˆͺ (πΉβ€˜π‘›))
6563, 64uneq12d 4125 . . . . . . . . . . . . 13 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑀 βˆͺ βˆͺ 𝑀) = ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)))
66 mpteq1 5199 . . . . . . . . . . . . . . . 16 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6766rneqd 5894 . . . . . . . . . . . . . . 15 (𝑀 = (πΉβ€˜π‘›) β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6867uneq2d 4124 . . . . . . . . . . . . . 14 (𝑀 = (πΉβ€˜π‘›) β†’ ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣})) = ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})))
6963, 68iuneq12d 4983 . . . . . . . . . . . . 13 (𝑀 = (πΉβ€˜π‘›) β†’ βˆͺ 𝑒 ∈ 𝑀 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣})) = βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})))
7065, 69uneq12d 4125 . . . . . . . . . . . 12 (𝑀 = (πΉβ€˜π‘›) β†’ ((𝑀 βˆͺ βˆͺ 𝑀) βˆͺ βˆͺ 𝑒 ∈ 𝑀 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}))) = (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))))
711, 62, 70frsucmpt2 8387 . . . . . . . . . . 11 ((𝑛 ∈ Ο‰ ∧ (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))) ∈ V) β†’ (πΉβ€˜suc 𝑛) = (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))))
7233, 42, 71sylancl 587 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜suc 𝑛) = (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))))
73 simpr 486 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
7427ad3antrrr 729 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
7573sselda 3945 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 ∈ (wUniClβ€˜π΄))
7674, 75wunelss 10649 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 βŠ† (wUniClβ€˜π΄))
7776ralrimiva 3140 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆ€π‘’ ∈ (πΉβ€˜π‘›)𝑒 βŠ† (wUniClβ€˜π΄))
78 unissb 4901 . . . . . . . . . . . . 13 (βˆͺ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄) ↔ βˆ€π‘’ ∈ (πΉβ€˜π‘›)𝑒 βŠ† (wUniClβ€˜π΄))
7977, 78sylibr 233 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆͺ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
8073, 79unssd 4147 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βŠ† (wUniClβ€˜π΄))
8174, 75wunpw 10648 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝒫 𝑒 ∈ (wUniClβ€˜π΄))
8274, 75wununi 10647 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ βˆͺ 𝑒 ∈ (wUniClβ€˜π΄))
8381, 82prssd 4783 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ {𝒫 𝑒, βˆͺ 𝑒} βŠ† (wUniClβ€˜π΄))
8474adantr 482 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
8575adantr 482 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 ∈ (wUniClβ€˜π΄))
86 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
8786sselda 3945 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ 𝑣 ∈ (wUniClβ€˜π΄))
8884, 85, 87wunpr 10650 . . . . . . . . . . . . . . . 16 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ {𝑒, 𝑣} ∈ (wUniClβ€˜π΄))
8988fmpttd 7064 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}):(πΉβ€˜π‘›)⟢(wUniClβ€˜π΄))
9089frnd 6677 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) βŠ† (wUniClβ€˜π΄))
9183, 90unssd 4147 . . . . . . . . . . . . 13 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) βŠ† (wUniClβ€˜π΄))
9291ralrimiva 3140 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆ€π‘’ ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) βŠ† (wUniClβ€˜π΄))
93 iunss 5006 . . . . . . . . . . . 12 (βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) βŠ† (wUniClβ€˜π΄) ↔ βˆ€π‘’ ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) βŠ† (wUniClβ€˜π΄))
9492, 93sylibr 233 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) βŠ† (wUniClβ€˜π΄))
9580, 94unssd 4147 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) βˆͺ βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))) βŠ† (wUniClβ€˜π΄))
9672, 95eqsstrd 3983 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄))
9796ex 414 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
9897expcom 415 . . . . . . 7 (𝑛 ∈ Ο‰ β†’ (𝐴 ∈ 𝑉 β†’ ((πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄))))
9913, 15, 17, 32, 98finds2 7838 . . . . . 6 (π‘š ∈ Ο‰ β†’ (𝐴 ∈ 𝑉 β†’ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄)))
10099com12 32 . . . . 5 (𝐴 ∈ 𝑉 β†’ (π‘š ∈ Ο‰ β†’ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄)))
101100ralrimiv 3139 . . . 4 (𝐴 ∈ 𝑉 β†’ βˆ€π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄))
102 iunss 5006 . . . 4 (βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ βˆ€π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄))
103101, 102sylibr 233 . . 3 (𝐴 ∈ 𝑉 β†’ βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄))
10411, 103eqsstrid 3993 . 2 (𝐴 ∈ 𝑉 β†’ π‘ˆ βŠ† (wUniClβ€˜π΄))
1055, 104eqssd 3962 1 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  {csn 4587  {cpr 4589  βˆͺ cuni 4866  βˆͺ ciun 4955   ↦ cmpt 5189  ran crn 5635   β†Ύ cres 5636  Oncon0 6318  suc csuc 6320   Fn wfn 6492  β€˜cfv 6497  Ο‰com 7803  reccrdg 8356  1oc1o 8406  WUnicwun 10641  wUniClcwunm 10642
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-wun 10643  df-wunc 10644
This theorem is referenced by: (None)
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