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Theorem wuncval2 10744
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 10735 . . 3 (𝐴 ∈ 𝑉 β†’ (π‘ˆ ∈ WUni ∧ 𝐴 βŠ† π‘ˆ))
4 wuncss 10742 . . 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 8437 . . . . . 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 6645 . . . . . 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 7248 . . . . 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 2761 . . 3 π‘ˆ = βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š)
12 fveq2 6890 . . . . . . . 8 (π‘š = βˆ… β†’ (πΉβ€˜π‘š) = (πΉβ€˜βˆ…))
1312sseq1d 4012 . . . . . . 7 (π‘š = βˆ… β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄)))
14 fveq2 6890 . . . . . . . 8 (π‘š = 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜π‘›))
1514sseq1d 4012 . . . . . . 7 (π‘š = 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)))
16 fveq2 6890 . . . . . . . 8 (π‘š = suc 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜suc 𝑛))
1716sseq1d 4012 . . . . . . 7 (π‘š = suc 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
18 1on 8480 . . . . . . . . . 10 1o ∈ On
19 unexg 7738 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) β†’ (𝐴 βˆͺ 1o) ∈ V)
2018, 19mpan2 687 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) ∈ V)
211fveq1i 6891 . . . . . . . . . 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 8438 . . . . . . . . . 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 2782 . . . . . . . . 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 10740 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 𝐴 βŠ† (wUniClβ€˜π΄))
26 df1o2 8475 . . . . . . . . . 10 1o = {βˆ…}
27 wunccl 10741 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) ∈ WUni)
2827wun0 10715 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ βˆ… ∈ (wUniClβ€˜π΄))
2928snssd 4811 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ {βˆ…} βŠ† (wUniClβ€˜π΄))
3026, 29eqsstrid 4029 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 1o βŠ† (wUniClβ€˜π΄))
3125, 30unssd 4185 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) βŠ† (wUniClβ€˜π΄))
3224, 31eqsstrd 4019 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄))
33 simplr 765 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ 𝑛 ∈ Ο‰)
34 fvex 6903 . . . . . . . . . . . . 13 (πΉβ€˜π‘›) ∈ V
3534uniex 7733 . . . . . . . . . . . . 13 βˆͺ (πΉβ€˜π‘›) ∈ V
3634, 35unex 7735 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) ∈ V
37 prex 5431 . . . . . . . . . . . . . 14 {𝒫 𝑒, βˆͺ 𝑒} ∈ V
3834mptex 7226 . . . . . . . . . . . . . . 15 (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
3938rnex 7905 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
4037, 39unex 7735 . . . . . . . . . . . . 13 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4134, 40iunex 7957 . . . . . . . . . . . 12 βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4236, 41unex 7735 . . . . . . . . . . 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 4918 . . . . . . . . . . . . . 14 (𝑀 = 𝑧 β†’ βˆͺ 𝑀 = βˆͺ 𝑧)
4543, 44uneq12d 4163 . . . . . . . . . . . . 13 (𝑀 = 𝑧 β†’ (𝑀 βˆͺ βˆͺ 𝑀) = (𝑧 βˆͺ βˆͺ 𝑧))
46 pweq 4615 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ 𝒫 𝑒 = 𝒫 π‘₯)
47 unieq 4918 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ βˆͺ 𝑒 = βˆͺ π‘₯)
4846, 47preq12d 4744 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ {𝒫 𝑒, βˆͺ 𝑒} = {𝒫 π‘₯, βˆͺ π‘₯})
49 preq1 4736 . . . . . . . . . . . . . . . . . 18 (𝑒 = π‘₯ β†’ {𝑒, 𝑣} = {π‘₯, 𝑣})
5049mpteq2dv 5249 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5150rneqd 5936 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5248, 51uneq12d 4163 . . . . . . . . . . . . . . 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 5042 . . . . . . . . . . . . . 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 4737 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 β†’ {π‘₯, 𝑣} = {π‘₯, 𝑦})
5554cbvmptv 5260 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦})
56 mpteq1 5240 . . . . . . . . . . . . . . . . . 18 (𝑀 = 𝑧 β†’ (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5755, 56eqtrid 2782 . . . . . . . . . . . . . . . . 17 (𝑀 = 𝑧 β†’ (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5857rneqd 5936 . . . . . . . . . . . . . . . 16 (𝑀 = 𝑧 β†’ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5958uneq2d 4162 . . . . . . . . . . . . . . 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 5024 . . . . . . . . . . . . . 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 2782 . . . . . . . . . . . . 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 4163 . . . . . . . . . . . 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 4918 . . . . . . . . . . . . . 14 (𝑀 = (πΉβ€˜π‘›) β†’ βˆͺ 𝑀 = βˆͺ (πΉβ€˜π‘›))
6563, 64uneq12d 4163 . . . . . . . . . . . . 13 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑀 βˆͺ βˆͺ 𝑀) = ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)))
66 mpteq1 5240 . . . . . . . . . . . . . . . 16 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6766rneqd 5936 . . . . . . . . . . . . . . 15 (𝑀 = (πΉβ€˜π‘›) β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6867uneq2d 4162 . . . . . . . . . . . . . 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 5024 . . . . . . . . . . . . 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 4163 . . . . . . . . . . . 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 8442 . . . . . . . . . . 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 584 . . . . . . . . . 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 483 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
7427ad3antrrr 726 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
7573sselda 3981 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 ∈ (wUniClβ€˜π΄))
7674, 75wunelss 10705 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 βŠ† (wUniClβ€˜π΄))
7776ralrimiva 3144 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆ€π‘’ ∈ (πΉβ€˜π‘›)𝑒 βŠ† (wUniClβ€˜π΄))
78 unissb 4942 . . . . . . . . . . . . 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 4185 . . . . . . . . . . 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 10704 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝒫 𝑒 ∈ (wUniClβ€˜π΄))
8274, 75wununi 10703 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ βˆͺ 𝑒 ∈ (wUniClβ€˜π΄))
8381, 82prssd 4824 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ {𝒫 𝑒, βˆͺ 𝑒} βŠ† (wUniClβ€˜π΄))
8474adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
8575adantr 479 . . . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
8786sselda 3981 . . . . . . . . . . . . . . . . 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 10706 . . . . . . . . . . . . . . . 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 7115 . . . . . . . . . . . . . . 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 6724 . . . . . . . . . . . . . 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 4185 . . . . . . . . . . . . 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 3144 . . . . . . . . . . . 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 5047 . . . . . . . . . . . 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 4185 . . . . . . . . . 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 4019 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄))
9796ex 411 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
9897expcom 412 . . . . . . 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 7893 . . . . . 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 3143 . . . 4 (𝐴 ∈ 𝑉 β†’ βˆ€π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄))
102 iunss 5047 . . . 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 4029 . 2 (𝐴 ∈ 𝑉 β†’ π‘ˆ βŠ† (wUniClβ€˜π΄))
1055, 104eqssd 3998 1 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  {cpr 4629  βˆͺ cuni 4907  βˆͺ ciun 4996   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677  Oncon0 6363  suc csuc 6365   Fn wfn 6537  β€˜cfv 6542  Ο‰com 7857  reccrdg 8411  1oc1o 8461  WUnicwun 10697  wUniClcwunm 10698
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-wun 10699  df-wunc 10700
This theorem is referenced by: (None)
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