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Theorem wuncval2 10746
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 10737 . . 3 (𝐴 ∈ 𝑉 β†’ (π‘ˆ ∈ WUni ∧ 𝐴 βŠ† π‘ˆ))
4 wuncss 10744 . . 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 8439 . . . . . 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 6646 . . . . . 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 7249 . . . . 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 2762 . . 3 π‘ˆ = βˆͺ π‘š ∈ Ο‰ (πΉβ€˜π‘š)
12 fveq2 6891 . . . . . . . 8 (π‘š = βˆ… β†’ (πΉβ€˜π‘š) = (πΉβ€˜βˆ…))
1312sseq1d 4013 . . . . . . 7 (π‘š = βˆ… β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄)))
14 fveq2 6891 . . . . . . . 8 (π‘š = 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜π‘›))
1514sseq1d 4013 . . . . . . 7 (π‘š = 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)))
16 fveq2 6891 . . . . . . . 8 (π‘š = suc 𝑛 β†’ (πΉβ€˜π‘š) = (πΉβ€˜suc 𝑛))
1716sseq1d 4013 . . . . . . 7 (π‘š = suc 𝑛 β†’ ((πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄) ↔ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
18 1on 8482 . . . . . . . . . 10 1o ∈ On
19 unexg 7740 . . . . . . . . . 10 ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) β†’ (𝐴 βˆͺ 1o) ∈ V)
2018, 19mpan2 688 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) ∈ V)
211fveq1i 6892 . . . . . . . . . 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 8440 . . . . . . . . . 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 2783 . . . . . . . . 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 10742 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 𝐴 βŠ† (wUniClβ€˜π΄))
26 df1o2 8477 . . . . . . . . . 10 1o = {βˆ…}
27 wunccl 10743 . . . . . . . . . . . 12 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) ∈ WUni)
2827wun0 10717 . . . . . . . . . . 11 (𝐴 ∈ 𝑉 β†’ βˆ… ∈ (wUniClβ€˜π΄))
2928snssd 4812 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ {βˆ…} βŠ† (wUniClβ€˜π΄))
3026, 29eqsstrid 4030 . . . . . . . . 9 (𝐴 ∈ 𝑉 β†’ 1o βŠ† (wUniClβ€˜π΄))
3125, 30unssd 4186 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ (𝐴 βˆͺ 1o) βŠ† (wUniClβ€˜π΄))
3224, 31eqsstrd 4020 . . . . . . 7 (𝐴 ∈ 𝑉 β†’ (πΉβ€˜βˆ…) βŠ† (wUniClβ€˜π΄))
33 simplr 766 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ 𝑛 ∈ Ο‰)
34 fvex 6904 . . . . . . . . . . . . 13 (πΉβ€˜π‘›) ∈ V
3534uniex 7735 . . . . . . . . . . . . 13 βˆͺ (πΉβ€˜π‘›) ∈ V
3634, 35unex 7737 . . . . . . . . . . . 12 ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)) ∈ V
37 prex 5432 . . . . . . . . . . . . . 14 {𝒫 𝑒, βˆͺ 𝑒} ∈ V
3834mptex 7227 . . . . . . . . . . . . . . 15 (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
3938rnex 7907 . . . . . . . . . . . . . 14 ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}) ∈ V
4037, 39unex 7737 . . . . . . . . . . . . 13 ({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4134, 40iunex 7959 . . . . . . . . . . . 12 βˆͺ 𝑒 ∈ (πΉβ€˜π‘›)({𝒫 𝑒, βˆͺ 𝑒} βˆͺ ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣})) ∈ V
4236, 41unex 7737 . . . . . . . . . . 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 4919 . . . . . . . . . . . . . 14 (𝑀 = 𝑧 β†’ βˆͺ 𝑀 = βˆͺ 𝑧)
4543, 44uneq12d 4164 . . . . . . . . . . . . 13 (𝑀 = 𝑧 β†’ (𝑀 βˆͺ βˆͺ 𝑀) = (𝑧 βˆͺ βˆͺ 𝑧))
46 pweq 4616 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ 𝒫 𝑒 = 𝒫 π‘₯)
47 unieq 4919 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ βˆͺ 𝑒 = βˆͺ π‘₯)
4846, 47preq12d 4745 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ {𝒫 𝑒, βˆͺ 𝑒} = {𝒫 π‘₯, βˆͺ π‘₯})
49 preq1 4737 . . . . . . . . . . . . . . . . . 18 (𝑒 = π‘₯ β†’ {𝑒, 𝑣} = {π‘₯, 𝑣})
5049mpteq2dv 5250 . . . . . . . . . . . . . . . . 17 (𝑒 = π‘₯ β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5150rneqd 5937 . . . . . . . . . . . . . . . 16 (𝑒 = π‘₯ β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}))
5248, 51uneq12d 4164 . . . . . . . . . . . . . . 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 5043 . . . . . . . . . . . . . 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 4738 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑦 β†’ {π‘₯, 𝑣} = {π‘₯, 𝑦})
5554cbvmptv 5261 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦})
56 mpteq1 5241 . . . . . . . . . . . . . . . . . 18 (𝑀 = 𝑧 β†’ (𝑦 ∈ 𝑀 ↦ {π‘₯, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5755, 56eqtrid 2783 . . . . . . . . . . . . . . . . 17 (𝑀 = 𝑧 β†’ (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5857rneqd 5937 . . . . . . . . . . . . . . . 16 (𝑀 = 𝑧 β†’ ran (𝑣 ∈ 𝑀 ↦ {π‘₯, 𝑣}) = ran (𝑦 ∈ 𝑧 ↦ {π‘₯, 𝑦}))
5958uneq2d 4163 . . . . . . . . . . . . . . 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 5025 . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . 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 4164 . . . . . . . . . . . 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 4919 . . . . . . . . . . . . . 14 (𝑀 = (πΉβ€˜π‘›) β†’ βˆͺ 𝑀 = βˆͺ (πΉβ€˜π‘›))
6563, 64uneq12d 4164 . . . . . . . . . . . . 13 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑀 βˆͺ βˆͺ 𝑀) = ((πΉβ€˜π‘›) βˆͺ βˆͺ (πΉβ€˜π‘›)))
66 mpteq1 5241 . . . . . . . . . . . . . . . 16 (𝑀 = (πΉβ€˜π‘›) β†’ (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6766rneqd 5937 . . . . . . . . . . . . . . 15 (𝑀 = (πΉβ€˜π‘›) β†’ ran (𝑣 ∈ 𝑀 ↦ {𝑒, 𝑣}) = ran (𝑣 ∈ (πΉβ€˜π‘›) ↦ {𝑒, 𝑣}))
6867uneq2d 4163 . . . . . . . . . . . . . 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 5025 . . . . . . . . . . . . 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 4164 . . . . . . . . . . . 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 8444 . . . . . . . . . . 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 585 . . . . . . . . . 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 484 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
7427ad3antrrr 727 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
7573sselda 3982 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 ∈ (wUniClβ€˜π΄))
7674, 75wunelss 10707 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝑒 βŠ† (wUniClβ€˜π΄))
7776ralrimiva 3145 . . . . . . . . . . . . 13 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ βˆ€π‘’ ∈ (πΉβ€˜π‘›)𝑒 βŠ† (wUniClβ€˜π΄))
78 unissb 4943 . . . . . . . . . . . . 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 4186 . . . . . . . . . . 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 10706 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ 𝒫 𝑒 ∈ (wUniClβ€˜π΄))
8274, 75wununi 10705 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ βˆͺ 𝑒 ∈ (wUniClβ€˜π΄))
8381, 82prssd 4825 . . . . . . . . . . . . . 14 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ {𝒫 𝑒, βˆͺ 𝑒} βŠ† (wUniClβ€˜π΄))
8474adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) ∧ 𝑣 ∈ (πΉβ€˜π‘›)) β†’ (wUniClβ€˜π΄) ∈ WUni)
8575adantr 480 . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) ∧ 𝑒 ∈ (πΉβ€˜π‘›)) β†’ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄))
8786sselda 3982 . . . . . . . . . . . . . . . . 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 10708 . . . . . . . . . . . . . . . 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 7116 . . . . . . . . . . . . . . 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 6725 . . . . . . . . . . . . . 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 4186 . . . . . . . . . . . . 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 3145 . . . . . . . . . . . 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 5048 . . . . . . . . . . . 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 4186 . . . . . . . . . 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 4020 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) ∧ (πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄)) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄))
9796ex 412 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝑛 ∈ Ο‰) β†’ ((πΉβ€˜π‘›) βŠ† (wUniClβ€˜π΄) β†’ (πΉβ€˜suc 𝑛) βŠ† (wUniClβ€˜π΄)))
9897expcom 413 . . . . . . 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 7895 . . . . . 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 3144 . . . 4 (𝐴 ∈ 𝑉 β†’ βˆ€π‘š ∈ Ο‰ (πΉβ€˜π‘š) βŠ† (wUniClβ€˜π΄))
102 iunss 5048 . . . 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 4030 . 2 (𝐴 ∈ 𝑉 β†’ π‘ˆ βŠ† (wUniClβ€˜π΄))
1055, 104eqssd 3999 1 (𝐴 ∈ 𝑉 β†’ (wUniClβ€˜π΄) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  Vcvv 3473   βˆͺ cun 3946   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {csn 4628  {cpr 4630  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678  Oncon0 6364  suc csuc 6366   Fn wfn 6538  β€˜cfv 6543  Ο‰com 7859  reccrdg 8413  1oc1o 8463  WUnicwun 10699  wUniClcwunm 10700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-wun 10701  df-wunc 10702
This theorem is referenced by: (None)
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