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Theorem stoweidlem27 44342
Description: This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here (π‘žβ€˜π‘–) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem27.1 𝐺 = (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
stoweidlem27.2 (πœ‘ β†’ 𝑄 ∈ V)
stoweidlem27.3 (πœ‘ β†’ 𝑀 ∈ β„•)
stoweidlem27.4 (πœ‘ β†’ π‘Œ Fn ran 𝐺)
stoweidlem27.5 (πœ‘ β†’ ran 𝐺 ∈ V)
stoweidlem27.6 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ (π‘Œβ€˜π‘™) ∈ 𝑙)
stoweidlem27.7 (πœ‘ β†’ 𝐹:(1...𝑀)–1-1-ontoβ†’ran 𝐺)
stoweidlem27.8 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋)
stoweidlem27.9 β„²π‘‘πœ‘
stoweidlem27.10 β„²π‘€πœ‘
stoweidlem27.11 β„²β„Žπ‘„
Assertion
Ref Expression
stoweidlem27 (πœ‘ β†’ βˆƒπ‘ž(𝑀 ∈ β„• ∧ (π‘ž:(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
Distinct variable groups:   β„Ž,𝑖,𝑑,𝑀,𝐹   β„Ž,𝑙,π‘Œ,𝑑,𝑀   𝑇,β„Ž,𝑀   𝑖,π‘ž,𝑑,𝐹   𝑖,𝐺   𝑖,𝑀,π‘ž   𝑖,𝑋,𝑀   𝑖,π‘Œ,π‘ž   πœ‘,𝑖   𝑄,𝑙   πœ‘,𝑙   𝐺,𝑙   𝑄,π‘ž   𝑇,π‘ž   π‘ˆ,π‘ž   𝑀,𝑀   𝑀,𝑄   𝑀,π‘ˆ
Allowed substitution hints:   πœ‘(𝑀,𝑑,β„Ž,π‘ž)   𝑄(𝑑,β„Ž,𝑖)   𝑇(𝑑,𝑖,𝑙)   π‘ˆ(𝑑,β„Ž,𝑖,𝑙)   𝐹(𝑙)   𝐺(𝑀,𝑑,β„Ž,π‘ž)   𝑀(𝑑,β„Ž,𝑙)   𝑋(𝑑,β„Ž,π‘ž,𝑙)

Proof of Theorem stoweidlem27
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 stoweidlem27.4 . . . 4 (πœ‘ β†’ π‘Œ Fn ran 𝐺)
2 stoweidlem27.5 . . . 4 (πœ‘ β†’ ran 𝐺 ∈ V)
3 fnex 7172 . . . 4 ((π‘Œ Fn ran 𝐺 ∧ ran 𝐺 ∈ V) β†’ π‘Œ ∈ V)
41, 2, 3syl2anc 585 . . 3 (πœ‘ β†’ π‘Œ ∈ V)
5 stoweidlem27.7 . . . . 5 (πœ‘ β†’ 𝐹:(1...𝑀)–1-1-ontoβ†’ran 𝐺)
6 f1ofn 6790 . . . . 5 (𝐹:(1...𝑀)–1-1-ontoβ†’ran 𝐺 β†’ 𝐹 Fn (1...𝑀))
75, 6syl 17 . . . 4 (πœ‘ β†’ 𝐹 Fn (1...𝑀))
8 ovex 7395 . . . 4 (1...𝑀) ∈ V
9 fnex 7172 . . . 4 ((𝐹 Fn (1...𝑀) ∧ (1...𝑀) ∈ V) β†’ 𝐹 ∈ V)
107, 8, 9sylancl 587 . . 3 (πœ‘ β†’ 𝐹 ∈ V)
11 coexg 7871 . . 3 ((π‘Œ ∈ V ∧ 𝐹 ∈ V) β†’ (π‘Œ ∘ 𝐹) ∈ V)
124, 10, 11syl2anc 585 . 2 (πœ‘ β†’ (π‘Œ ∘ 𝐹) ∈ V)
13 stoweidlem27.3 . . 3 (πœ‘ β†’ 𝑀 ∈ β„•)
14 f1of 6789 . . . . . 6 (𝐹:(1...𝑀)–1-1-ontoβ†’ran 𝐺 β†’ 𝐹:(1...𝑀)⟢ran 𝐺)
155, 14syl 17 . . . . 5 (πœ‘ β†’ 𝐹:(1...𝑀)⟢ran 𝐺)
16 fnfco 6712 . . . . 5 ((π‘Œ Fn ran 𝐺 ∧ 𝐹:(1...𝑀)⟢ran 𝐺) β†’ (π‘Œ ∘ 𝐹) Fn (1...𝑀))
171, 15, 16syl2anc 585 . . . 4 (πœ‘ β†’ (π‘Œ ∘ 𝐹) Fn (1...𝑀))
18 rncoss 5932 . . . . 5 ran (π‘Œ ∘ 𝐹) βŠ† ran π‘Œ
19 fvelrnb 6908 . . . . . . . . . . 11 (π‘Œ Fn ran 𝐺 β†’ (π‘˜ ∈ ran π‘Œ ↔ βˆƒπ‘™ ∈ ran 𝐺(π‘Œβ€˜π‘™) = π‘˜))
201, 19syl 17 . . . . . . . . . 10 (πœ‘ β†’ (π‘˜ ∈ ran π‘Œ ↔ βˆƒπ‘™ ∈ ran 𝐺(π‘Œβ€˜π‘™) = π‘˜))
2120biimpa 478 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ βˆƒπ‘™ ∈ ran 𝐺(π‘Œβ€˜π‘™) = π‘˜)
22 stoweidlem27.10 . . . . . . . . . . . . . 14 β„²π‘€πœ‘
23 stoweidlem27.1 . . . . . . . . . . . . . . . . 17 𝐺 = (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
24 nfmpt1 5218 . . . . . . . . . . . . . . . . 17 Ⅎ𝑀(𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
2523, 24nfcxfr 2906 . . . . . . . . . . . . . . . 16 Ⅎ𝑀𝐺
2625nfrn 5912 . . . . . . . . . . . . . . 15 Ⅎ𝑀ran 𝐺
2726nfcri 2895 . . . . . . . . . . . . . 14 Ⅎ𝑀 𝑙 ∈ ran 𝐺
2822, 27nfan 1903 . . . . . . . . . . . . 13 Ⅎ𝑀(πœ‘ ∧ 𝑙 ∈ ran 𝐺)
29 stoweidlem27.6 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ (π‘Œβ€˜π‘™) ∈ 𝑙)
3029ad2antrr 725 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ (π‘Œβ€˜π‘™) ∈ 𝑙)
31 simpr 486 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
3230, 31eleqtrd 2840 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ (π‘Œβ€˜π‘™) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
33 nfcv 2908 . . . . . . . . . . . . . . . 16 β„²β„Ž(π‘Œβ€˜π‘™)
34 stoweidlem27.11 . . . . . . . . . . . . . . . 16 β„²β„Žπ‘„
35 nfv 1918 . . . . . . . . . . . . . . . 16 β„²β„Ž 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)}
36 fveq1 6846 . . . . . . . . . . . . . . . . . . 19 (β„Ž = (π‘Œβ€˜π‘™) β†’ (β„Žβ€˜π‘‘) = ((π‘Œβ€˜π‘™)β€˜π‘‘))
3736breq2d 5122 . . . . . . . . . . . . . . . . . 18 (β„Ž = (π‘Œβ€˜π‘™) β†’ (0 < (β„Žβ€˜π‘‘) ↔ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)))
3837rabbidv 3418 . . . . . . . . . . . . . . . . 17 (β„Ž = (π‘Œβ€˜π‘™) β†’ {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)} = {𝑑 ∈ 𝑇 ∣ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)})
3938eqeq2d 2748 . . . . . . . . . . . . . . . 16 (β„Ž = (π‘Œβ€˜π‘™) β†’ (𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)} ↔ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)}))
4033, 34, 35, 39elrabf 3646 . . . . . . . . . . . . . . 15 ((π‘Œβ€˜π‘™) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ↔ ((π‘Œβ€˜π‘™) ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)}))
4132, 40sylib 217 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ ((π‘Œβ€˜π‘™) ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < ((π‘Œβ€˜π‘™)β€˜π‘‘)}))
4241simpld 496 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ∧ 𝑀 ∈ 𝑋) ∧ 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) β†’ (π‘Œβ€˜π‘™) ∈ 𝑄)
43 simpr 486 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ 𝑙 ∈ ran 𝐺)
4423elrnmpt 5916 . . . . . . . . . . . . . . 15 (𝑙 ∈ ran 𝐺 β†’ (𝑙 ∈ ran 𝐺 ↔ βˆƒπ‘€ ∈ 𝑋 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}))
4543, 44syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ (𝑙 ∈ ran 𝐺 ↔ βˆƒπ‘€ ∈ 𝑋 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}))
4643, 45mpbid 231 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ βˆƒπ‘€ ∈ 𝑋 𝑙 = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
4728, 42, 46r19.29af 3254 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ (π‘Œβ€˜π‘™) ∈ 𝑄)
4847adantlr 714 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) ∧ 𝑙 ∈ ran 𝐺) β†’ (π‘Œβ€˜π‘™) ∈ 𝑄)
49 eleq1 2826 . . . . . . . . . . 11 ((π‘Œβ€˜π‘™) = π‘˜ β†’ ((π‘Œβ€˜π‘™) ∈ 𝑄 ↔ π‘˜ ∈ 𝑄))
5048, 49syl5ibcom 244 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) ∧ 𝑙 ∈ ran 𝐺) β†’ ((π‘Œβ€˜π‘™) = π‘˜ β†’ π‘˜ ∈ 𝑄))
5150reximdva 3166 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ (βˆƒπ‘™ ∈ ran 𝐺(π‘Œβ€˜π‘™) = π‘˜ β†’ βˆƒπ‘™ ∈ ran 𝐺 π‘˜ ∈ 𝑄))
5221, 51mpd 15 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ βˆƒπ‘™ ∈ ran 𝐺 π‘˜ ∈ 𝑄)
53 idd 24 . . . . . . . . . 10 (𝑙 ∈ ran 𝐺 β†’ (π‘˜ ∈ 𝑄 β†’ π‘˜ ∈ 𝑄))
5453a1i 11 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ (𝑙 ∈ ran 𝐺 β†’ (π‘˜ ∈ 𝑄 β†’ π‘˜ ∈ 𝑄)))
5554rexlimdv 3151 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ (βˆƒπ‘™ ∈ ran 𝐺 π‘˜ ∈ 𝑄 β†’ π‘˜ ∈ 𝑄))
5652, 55mpd 15 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ ran π‘Œ) β†’ π‘˜ ∈ 𝑄)
5756ex 414 . . . . . 6 (πœ‘ β†’ (π‘˜ ∈ ran π‘Œ β†’ π‘˜ ∈ 𝑄))
5857ssrdv 3955 . . . . 5 (πœ‘ β†’ ran π‘Œ βŠ† 𝑄)
5918, 58sstrid 3960 . . . 4 (πœ‘ β†’ ran (π‘Œ ∘ 𝐹) βŠ† 𝑄)
60 df-f 6505 . . . 4 ((π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„ ↔ ((π‘Œ ∘ 𝐹) Fn (1...𝑀) ∧ ran (π‘Œ ∘ 𝐹) βŠ† 𝑄))
6117, 59, 60sylanbrc 584 . . 3 (πœ‘ β†’ (π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„)
62 stoweidlem27.9 . . . 4 β„²π‘‘πœ‘
63 nfv 1918 . . . . . . 7 Ⅎ𝑀 𝑑 ∈ (𝑇 βˆ– π‘ˆ)
6422, 63nfan 1903 . . . . . 6 Ⅎ𝑀(πœ‘ ∧ 𝑑 ∈ (𝑇 βˆ– π‘ˆ))
65 nfv 1918 . . . . . 6 β„²π‘€βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)
66 stoweidlem27.8 . . . . . . . 8 (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑋)
6766sselda 3949 . . . . . . 7 ((πœ‘ ∧ 𝑑 ∈ (𝑇 βˆ– π‘ˆ)) β†’ 𝑑 ∈ βˆͺ 𝑋)
68 eluni 4873 . . . . . . 7 (𝑑 ∈ βˆͺ 𝑋 ↔ βˆƒπ‘€(𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋))
6967, 68sylib 217 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ (𝑇 βˆ– π‘ˆ)) β†’ βˆƒπ‘€(𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋))
7023funmpt2 6545 . . . . . . . . . . . 12 Fun 𝐺
7123dmeqi 5865 . . . . . . . . . . . . . . 15 dom 𝐺 = dom (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
72 stoweidlem27.2 . . . . . . . . . . . . . . . . . . . 20 (πœ‘ β†’ 𝑄 ∈ V)
7334rabexgf 43303 . . . . . . . . . . . . . . . . . . . 20 (𝑄 ∈ V β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
7472, 73syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
7574adantr 482 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
7675ex 414 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (𝑀 ∈ 𝑋 β†’ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V))
7722, 76ralrimi 3243 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ βˆ€π‘€ ∈ 𝑋 {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V)
78 dmmptg 6199 . . . . . . . . . . . . . . . 16 (βˆ€π‘€ ∈ 𝑋 {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V β†’ dom (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) = 𝑋)
7977, 78syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ dom (𝑀 ∈ 𝑋 ↦ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}) = 𝑋)
8071, 79eqtrid 2789 . . . . . . . . . . . . . 14 (πœ‘ β†’ dom 𝐺 = 𝑋)
8180eleq2d 2824 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝑀 ∈ dom 𝐺 ↔ 𝑀 ∈ 𝑋))
8281biimpar 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ 𝑀 ∈ dom 𝐺)
83 fvelrn 7032 . . . . . . . . . . . 12 ((Fun 𝐺 ∧ 𝑀 ∈ dom 𝐺) β†’ (πΊβ€˜π‘€) ∈ ran 𝐺)
8470, 82, 83sylancr 588 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ (πΊβ€˜π‘€) ∈ ran 𝐺)
8584adantrl 715 . . . . . . . . . 10 ((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) β†’ (πΊβ€˜π‘€) ∈ ran 𝐺)
8615ad2antrr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ 𝐹:(1...𝑀)⟢ran 𝐺)
87 simprl 770 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ 𝑖 ∈ (1...𝑀))
88 fvco3 6945 . . . . . . . . . . . . . 14 ((𝐹:(1...𝑀)⟢ran 𝐺 ∧ 𝑖 ∈ (1...𝑀)) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) = (π‘Œβ€˜(πΉβ€˜π‘–)))
8986, 87, 88syl2anc 585 . . . . . . . . . . . . 13 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) = (π‘Œβ€˜(πΉβ€˜π‘–)))
90 fveq2 6847 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘–) = (πΊβ€˜π‘€) β†’ (π‘Œβ€˜(πΉβ€˜π‘–)) = (π‘Œβ€˜(πΊβ€˜π‘€)))
9190ad2antll 728 . . . . . . . . . . . . 13 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ (π‘Œβ€˜(πΉβ€˜π‘–)) = (π‘Œβ€˜(πΊβ€˜π‘€)))
9289, 91eqtrd 2777 . . . . . . . . . . . 12 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) = (π‘Œβ€˜(πΊβ€˜π‘€)))
93 eleq1 2826 . . . . . . . . . . . . . . . . 17 (𝑙 = (πΊβ€˜π‘€) β†’ (𝑙 ∈ ran 𝐺 ↔ (πΊβ€˜π‘€) ∈ ran 𝐺))
9493anbi2d 630 . . . . . . . . . . . . . . . 16 (𝑙 = (πΊβ€˜π‘€) β†’ ((πœ‘ ∧ 𝑙 ∈ ran 𝐺) ↔ (πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺)))
95 eleq2 2827 . . . . . . . . . . . . . . . . 17 (𝑙 = (πΊβ€˜π‘€) β†’ ((π‘Œβ€˜π‘™) ∈ 𝑙 ↔ (π‘Œβ€˜π‘™) ∈ (πΊβ€˜π‘€)))
96 fveq2 6847 . . . . . . . . . . . . . . . . . 18 (𝑙 = (πΊβ€˜π‘€) β†’ (π‘Œβ€˜π‘™) = (π‘Œβ€˜(πΊβ€˜π‘€)))
9796eleq1d 2823 . . . . . . . . . . . . . . . . 17 (𝑙 = (πΊβ€˜π‘€) β†’ ((π‘Œβ€˜π‘™) ∈ (πΊβ€˜π‘€) ↔ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€)))
9895, 97bitrd 279 . . . . . . . . . . . . . . . 16 (𝑙 = (πΊβ€˜π‘€) β†’ ((π‘Œβ€˜π‘™) ∈ 𝑙 ↔ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€)))
9994, 98imbi12d 345 . . . . . . . . . . . . . . 15 (𝑙 = (πΊβ€˜π‘€) β†’ (((πœ‘ ∧ 𝑙 ∈ ran 𝐺) β†’ (π‘Œβ€˜π‘™) ∈ 𝑙) ↔ ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€))))
10099, 29vtoclg 3528 . . . . . . . . . . . . . 14 ((πΊβ€˜π‘€) ∈ ran 𝐺 β†’ ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€)))
101100anabsi7 670 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€))
102101adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ (π‘Œβ€˜(πΊβ€˜π‘€)) ∈ (πΊβ€˜π‘€))
10392, 102eqeltrd 2838 . . . . . . . . . . 11 (((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) ∧ (𝑖 ∈ (1...𝑀) ∧ (πΉβ€˜π‘–) = (πΊβ€˜π‘€))) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€))
104 f1ofo 6796 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–1-1-ontoβ†’ran 𝐺 β†’ 𝐹:(1...𝑀)–ontoβ†’ran 𝐺)
105 forn 6764 . . . . . . . . . . . . . . 15 (𝐹:(1...𝑀)–ontoβ†’ran 𝐺 β†’ ran 𝐹 = ran 𝐺)
1065, 104, 1053syl 18 . . . . . . . . . . . . . 14 (πœ‘ β†’ ran 𝐹 = ran 𝐺)
107106eleq2d 2824 . . . . . . . . . . . . 13 (πœ‘ β†’ ((πΊβ€˜π‘€) ∈ ran 𝐹 ↔ (πΊβ€˜π‘€) ∈ ran 𝐺))
108107biimpar 479 . . . . . . . . . . . 12 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ (πΊβ€˜π‘€) ∈ ran 𝐹)
1097adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ 𝐹 Fn (1...𝑀))
110 fvelrnb 6908 . . . . . . . . . . . . 13 (𝐹 Fn (1...𝑀) β†’ ((πΊβ€˜π‘€) ∈ ran 𝐹 ↔ βˆƒπ‘– ∈ (1...𝑀)(πΉβ€˜π‘–) = (πΊβ€˜π‘€)))
111109, 110syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ ((πΊβ€˜π‘€) ∈ ran 𝐹 ↔ βˆƒπ‘– ∈ (1...𝑀)(πΉβ€˜π‘–) = (πΊβ€˜π‘€)))
112108, 111mpbid 231 . . . . . . . . . . 11 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ βˆƒπ‘– ∈ (1...𝑀)(πΉβ€˜π‘–) = (πΊβ€˜π‘€))
113103, 112reximddv 3169 . . . . . . . . . 10 ((πœ‘ ∧ (πΊβ€˜π‘€) ∈ ran 𝐺) β†’ βˆƒπ‘– ∈ (1...𝑀)((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€))
11485, 113syldan 592 . . . . . . . . 9 ((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) β†’ βˆƒπ‘– ∈ (1...𝑀)((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€))
115 simplrl 776 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ 𝑑 ∈ 𝑀)
116 simpr 486 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ 𝑀 ∈ 𝑋)
11723fvmpt2 6964 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 ∈ 𝑋 ∧ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ∈ V) β†’ (πΊβ€˜π‘€) = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
118116, 75, 117syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ (πΊβ€˜π‘€) = {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
119118eleq2d 2824 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑀 ∈ 𝑋) β†’ (((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€) ↔ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}))
120119biimpa 478 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑀 ∈ 𝑋) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
121120adantlrl 719 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}})
122 nfcv 2908 . . . . . . . . . . . . . . . . 17 β„²β„Ž((π‘Œ ∘ 𝐹)β€˜π‘–)
123 nfv 1918 . . . . . . . . . . . . . . . . 17 β„²β„Ž 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)}
124 fveq1 6846 . . . . . . . . . . . . . . . . . . . 20 (β„Ž = ((π‘Œ ∘ 𝐹)β€˜π‘–) β†’ (β„Žβ€˜π‘‘) = (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
125124breq2d 5122 . . . . . . . . . . . . . . . . . . 19 (β„Ž = ((π‘Œ ∘ 𝐹)β€˜π‘–) β†’ (0 < (β„Žβ€˜π‘‘) ↔ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
126125rabbidv 3418 . . . . . . . . . . . . . . . . . 18 (β„Ž = ((π‘Œ ∘ 𝐹)β€˜π‘–) β†’ {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)} = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)})
127126eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (β„Ž = ((π‘Œ ∘ 𝐹)β€˜π‘–) β†’ (𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)} ↔ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)}))
128122, 34, 123, 127elrabf 3646 . . . . . . . . . . . . . . . 16 (((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ {β„Ž ∈ 𝑄 ∣ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}} ↔ (((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)}))
129121, 128sylib 217 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ (((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ 𝑄 ∧ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)}))
130129simprd 497 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)})
131115, 130eleqtrd 2840 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ 𝑑 ∈ {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)})
132 rabid 3430 . . . . . . . . . . . . 13 (𝑑 ∈ {𝑑 ∈ 𝑇 ∣ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)} ↔ (𝑑 ∈ 𝑇 ∧ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
133131, 132sylib 217 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ (𝑑 ∈ 𝑇 ∧ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
134133simprd 497 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) ∧ ((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€)) β†’ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
135134ex 414 . . . . . . . . . 10 ((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) β†’ (((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€) β†’ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
136135reximdv 3168 . . . . . . . . 9 ((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) β†’ (βˆƒπ‘– ∈ (1...𝑀)((π‘Œ ∘ 𝐹)β€˜π‘–) ∈ (πΊβ€˜π‘€) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
137114, 136mpd 15 . . . . . . . 8 ((πœ‘ ∧ (𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋)) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
138137ex 414 . . . . . . 7 (πœ‘ β†’ ((𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
139138adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑑 ∈ (𝑇 βˆ– π‘ˆ)) β†’ ((𝑑 ∈ 𝑀 ∧ 𝑀 ∈ 𝑋) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
14064, 65, 69, 139exlimimdd 2213 . . . . 5 ((πœ‘ ∧ 𝑑 ∈ (𝑇 βˆ– π‘ˆ)) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
141140ex 414 . . . 4 (πœ‘ β†’ (𝑑 ∈ (𝑇 βˆ– π‘ˆ) β†’ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
14262, 141ralrimi 3243 . . 3 (πœ‘ β†’ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
14313, 61, 142jca32 517 . 2 (πœ‘ β†’ (𝑀 ∈ β„• ∧ ((π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))))
144 feq1 6654 . . . . 5 (π‘ž = (π‘Œ ∘ 𝐹) β†’ (π‘ž:(1...𝑀)βŸΆπ‘„ ↔ (π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„))
145 fveq1 6846 . . . . . . . . 9 (π‘ž = (π‘Œ ∘ 𝐹) β†’ (π‘žβ€˜π‘–) = ((π‘Œ ∘ 𝐹)β€˜π‘–))
146145fveq1d 6849 . . . . . . . 8 (π‘ž = (π‘Œ ∘ 𝐹) β†’ ((π‘žβ€˜π‘–)β€˜π‘‘) = (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))
147146breq2d 5122 . . . . . . 7 (π‘ž = (π‘Œ ∘ 𝐹) β†’ (0 < ((π‘žβ€˜π‘–)β€˜π‘‘) ↔ 0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
148147rexbidv 3176 . . . . . 6 (π‘ž = (π‘Œ ∘ 𝐹) β†’ (βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘) ↔ βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
149148ralbidv 3175 . . . . 5 (π‘ž = (π‘Œ ∘ 𝐹) β†’ (βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘) ↔ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))
150144, 149anbi12d 632 . . . 4 (π‘ž = (π‘Œ ∘ 𝐹) β†’ ((π‘ž:(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘)) ↔ ((π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))))
151150anbi2d 630 . . 3 (π‘ž = (π‘Œ ∘ 𝐹) β†’ ((𝑀 ∈ β„• ∧ (π‘ž:(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))) ↔ (𝑀 ∈ β„• ∧ ((π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘)))))
152151spcegv 3559 . 2 ((π‘Œ ∘ 𝐹) ∈ V β†’ ((𝑀 ∈ β„• ∧ ((π‘Œ ∘ 𝐹):(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < (((π‘Œ ∘ 𝐹)β€˜π‘–)β€˜π‘‘))) β†’ βˆƒπ‘ž(𝑀 ∈ β„• ∧ (π‘ž:(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘)))))
15312, 143, 152sylc 65 1 (πœ‘ β†’ βˆƒπ‘ž(𝑀 ∈ β„• ∧ (π‘ž:(1...𝑀)βŸΆπ‘„ ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)βˆƒπ‘– ∈ (1...𝑀)0 < ((π‘žβ€˜π‘–)β€˜π‘‘))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782  β„²wnf 1786   ∈ wcel 2107  β„²wnfc 2888  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  βˆͺ cuni 4870   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5638  ran crn 5639   ∘ ccom 5642  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  1c1 11059   < clt 11196  β„•cn 12160  ...cfz 13431
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365
This theorem is referenced by:  stoweidlem35  44350
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