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Theorem fin23lem30 9354
Description: Lemma for fin23 9401. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem30 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . 2 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4268 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
32biimpi 206 . 2 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
4 simpr 479 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5 fin23lem.d . . . . . . . . . . . 12 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
65funmpt2 6086 . . . . . . . . . . 11 Fun 𝑅
7 funco 6087 . . . . . . . . . . 11 ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡𝑅))
84, 6, 7sylancl 697 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡𝑅))
9 elunirn 6670 . . . . . . . . . 10 (Fun (𝑡𝑅) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
108, 9syl 17 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) ↔ ∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏)))
11 dmcoss 5538 . . . . . . . . . . . 12 dom (𝑡𝑅) ⊆ dom 𝑅
1211sseli 3738 . . . . . . . . . . 11 (𝑏 ∈ dom (𝑡𝑅) → 𝑏 ∈ dom 𝑅)
13 fvco 6434 . . . . . . . . . . . . . . . 16 ((Fun 𝑅𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
146, 13mpan 708 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑅 → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1514adantl 473 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1615eleq2d 2823 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅𝑏))))
17 incom 3946 . . . . . . . . . . . . . . . 16 ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏)))
18 difss 3878 . . . . . . . . . . . . . . . . . . . . . . 23 (ω ∖ 𝑃) ⊆ ω
19 ominf 8335 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ ω ∈ Fin
20 fin23lem.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
21 ssrab2 3826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ⊆ ω
2220, 21eqsstri 3774 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑃 ⊆ ω
23 undif 4191 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
2422, 23mpbi 220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 ∪ (ω ∖ 𝑃)) = ω
25 unfi 8390 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
2624, 25syl5eqelr 2842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
2726ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
2819, 27mtoi 190 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
2928ad2antrr 764 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
305fin23lem22 9339 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
3118, 29, 30sylancr 698 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
32 f1of 6296 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
3331, 32syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
34 simpr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
35 fdm 6210 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅:ω⟶(ω ∖ 𝑃) → dom 𝑅 = ω)
3633, 35syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
3734, 36eleqtrd 2839 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
3833, 37ffvelrnd 6521 . . . . . . . . . . . . . . . . . . . 20 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ (ω ∖ 𝑃))
3938eldifbd 3726 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ 𝑃)
4020eleq2i 2829 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ 𝑃 ↔ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4139, 40sylnib 317 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)})
4238eldifad 3725 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ ω)
43 fveq2 6350 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑅𝑏) → (𝑡𝑣) = (𝑡‘(𝑅𝑏)))
4443sseq2d 3772 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑅𝑏) → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4544elrab3 3503 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑏) ∈ ω → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4642, 45syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅𝑏) ∈ {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4741, 46mtbid 313 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)))
48 fin23lem.a . . . . . . . . . . . . . . . . . . 19 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
4948fin23lem20 9349 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑏) ∈ ω → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
5042, 49syl 17 . . . . . . . . . . . . . . . . 17 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
51 orel1 396 . . . . . . . . . . . . . . . . 17 ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) → (( ran 𝑈 ⊆ (𝑡‘(𝑅𝑏)) ∨ ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅))
5247, 50, 51sylc 65 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏))) = ∅)
5317, 52syl5eq 2804 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅)
54 disj 4158 . . . . . . . . . . . . . . 15 (((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
5553, 54sylib 208 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈)
56 rsp 3065 . . . . . . . . . . . . . 14 (∀𝑎 ∈ (𝑡‘(𝑅𝑏)) ¬ 𝑎 ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5755, 56syl 17 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅𝑏)) → ¬ 𝑎 ran 𝑈))
5816, 57sylbid 230 . . . . . . . . . . . 12 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
5958ex 449 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
6012, 59syl5 34 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈)))
6160rexlimdv 3166 . . . . . . . . 9 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡𝑅)𝑎 ∈ ((𝑡𝑅)‘𝑏) → ¬ 𝑎 ran 𝑈))
6210, 61sylbid 230 . . . . . . . 8 ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ran (𝑡𝑅) → ¬ 𝑎 ran 𝑈))
6362ralrimiv 3101 . . . . . . 7 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
64 disj 4158 . . . . . . 7 (( ran (𝑡𝑅) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
6563, 64sylibr 224 . . . . . 6 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅)
66 rneq 5504 . . . . . . . . 9 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6766unieqd 4596 . . . . . . . 8 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6867ineq1d 3954 . . . . . . 7 (𝑍 = (𝑡𝑅) → ( ran 𝑍 ran 𝑈) = ( ran (𝑡𝑅) ∩ ran 𝑈))
6968eqeq1d 2760 . . . . . 6 (𝑍 = (𝑡𝑅) → (( ran 𝑍 ran 𝑈) = ∅ ↔ ( ran (𝑡𝑅) ∩ ran 𝑈) = ∅))
7065, 69syl5ibr 236 . . . . 5 (𝑍 = (𝑡𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → ( ran 𝑍 ran 𝑈) = ∅))
7170expd 451 . . . 4 (𝑍 = (𝑡𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)))
7271impcom 445 . . 3 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
73 rneq 5504 . . . . . . . 8 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7473unieqd 4596 . . . . . . 7 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7574ineq1d 3954 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈))
76 rncoss 5539 . . . . . . . 8 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
7776unissi 4611 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
78 disj 4158 . . . . . . . 8 (( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ¬ 𝑎 ran 𝑈)
79 eluniab 4597 . . . . . . . . . 10 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)))
80 eleq2 2826 . . . . . . . . . . . . . 14 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈)))
81 eldifn 3874 . . . . . . . . . . . . . 14 (𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈) → ¬ 𝑎 ran 𝑈)
8280, 81syl6bi 243 . . . . . . . . . . . . 13 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8382rexlimivw 3165 . . . . . . . . . . . 12 (∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8483impcom 445 . . . . . . . . . . 11 ((𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8584exlimiv 2005 . . . . . . . . . 10 (∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8679, 85sylbi 207 . . . . . . . . 9 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} → ¬ 𝑎 ran 𝑈)
87 eqid 2758 . . . . . . . . . . 11 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
8887rnmpt 5524 . . . . . . . . . 10 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8988unieqi 4595 . . . . . . . . 9 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
9086, 89eleq2s 2855 . . . . . . . 8 (𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
9178, 90mprgbir 3063 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅
92 ssdisj 4168 . . . . . . 7 (( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∧ ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅) → ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅)
9377, 91, 92mp2an 710 . . . . . 6 ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅
9475, 93syl6eq 2808 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ∅)
9594a1d 25 . . . 4 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9695adantl 473 . . 3 ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
9772, 96jaoi 393 . 2 (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅))
981, 3, 97mp2b 10 1 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1630  wex 1851  wcel 2137  {cab 2744  wral 3048  wrex 3049  {crab 3052  Vcvv 3338  cdif 3710  cun 3711  cin 3712  wss 3713  c0 4056  ifcif 4228  𝒫 cpw 4300   cuni 4586   cint 4625   class class class wbr 4802  cmpt 4879  dom cdm 5264  ran crn 5265  ccom 5268  suc csuc 5884  Fun wfun 6041  wf 6043  1-1-ontowf1o 6046  cfv 6047  crio 6771  (class class class)co 6811  cmpt2 6813  ωcom 7228  seq𝜔cseqom 7709  𝑚 cmap 8021  cen 8116  Fincfn 8119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-seqom 7710  df-1o 7727  df-oadd 7731  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-card 8953
This theorem is referenced by:  fin23lem31  9355
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