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Theorem fin23lem30 9202
 Description: Lemma for fin23 9249. 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 4159 . . 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 476 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5 fin23lem.d . . . . . . . . . . . 12 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
65funmpt2 5965 . . . . . . . . . . 11 Fun 𝑅
7 funco 5966 . . . . . . . . . . 11 ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡𝑅))
84, 6, 7sylancl 695 . . . . . . . . . 10 ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡𝑅))
9 elunirn 6549 . . . . . . . . . 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 5417 . . . . . . . . . . . 12 dom (𝑡𝑅) ⊆ dom 𝑅
1211sseli 3632 . . . . . . . . . . 11 (𝑏 ∈ dom (𝑡𝑅) → 𝑏 ∈ dom 𝑅)
13 fvco 6313 . . . . . . . . . . . . . . . 16 ((Fun 𝑅𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
146, 13mpan 706 . . . . . . . . . . . . . . 15 (𝑏 ∈ dom 𝑅 → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1514adantl 481 . . . . . . . . . . . . . 14 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡𝑅)‘𝑏) = (𝑡‘(𝑅𝑏)))
1615eleq2d 2716 . . . . . . . . . . . . 13 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅𝑏))))
17 incom 3838 . . . . . . . . . . . . . . . 16 ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ( ran 𝑈 ∩ (𝑡‘(𝑅𝑏)))
18 difss 3770 . . . . . . . . . . . . . . . . . . . . . . 23 (ω ∖ 𝑃) ⊆ ω
19 ominf 8213 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ ω ∈ Fin
20 fin23lem.b . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
21 ssrab2 3720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)} ⊆ ω
2220, 21eqsstri 3668 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑃 ⊆ ω
23 undif 4082 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
2422, 23mpbi 220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃 ∪ (ω ∖ 𝑃)) = ω
25 unfi 8268 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
2624, 25syl5eqelr 2735 . . . . . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
305fin23lem22 9187 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
3118, 29, 30sylancr 696 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
32 f1of 6175 . . . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
35 fdm 6089 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅:ω⟶(ω ∖ 𝑃) → dom 𝑅 = ω)
3633, 35syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
3734, 36eleqtrd 2732 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
3833, 37ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . 20 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ (ω ∖ 𝑃))
3938eldifbd 3620 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅𝑏) ∈ 𝑃)
4020eleq2i 2722 . . . . . . . . . . . . . . . . . . 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 3619 . . . . . . . . . . . . . . . . . . 19 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅𝑏) ∈ ω)
43 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 = (𝑅𝑏) → (𝑡𝑣) = (𝑡‘(𝑅𝑏)))
4443sseq2d 3666 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑅𝑏) → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡‘(𝑅𝑏))))
4544elrab3 3397 . . . . . . . . . . . . . . . . . . 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 9197 . . . . . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . . . . 15 (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅𝑏)) ∩ ran 𝑈) = ∅)
54 disj 4050 . . . . . . . . . . . . . . 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 2958 . . . . . . . . . . . . . 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 3059 . . . . . . . . 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 2994 . . . . . . 7 ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ran (𝑡𝑅) ¬ 𝑎 ran 𝑈)
64 disj 4050 . . . . . . 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 5383 . . . . . . . . 9 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6766unieqd 4478 . . . . . . . 8 (𝑍 = (𝑡𝑅) → ran 𝑍 = ran (𝑡𝑅))
6867ineq1d 3846 . . . . . . 7 (𝑍 = (𝑡𝑅) → ( ran 𝑍 ran 𝑈) = ( ran (𝑡𝑅) ∩ ran 𝑈))
6968eqeq1d 2653 . . . . . 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 5383 . . . . . . . 8 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7473unieqd 4478 . . . . . . 7 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
7574ineq1d 3846 . . . . . 6 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → ( ran 𝑍 ran 𝑈) = ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈))
76 rncoss 5418 . . . . . . . 8 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
7776unissi 4493 . . . . . . 7 ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
78 disj 4050 . . . . . . . 8 (( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ¬ 𝑎 ran 𝑈)
79 eluniab 4479 . . . . . . . . . 10 (𝑎 {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)} ↔ ∃𝑏(𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)))
80 eleq2 2719 . . . . . . . . . . . . . 14 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈)))
81 eldifn 3766 . . . . . . . . . . . . . 14 (𝑎 ∈ ((𝑡𝑧) ∖ ran 𝑈) → ¬ 𝑎 ran 𝑈)
8280, 81syl6bi 243 . . . . . . . . . . . . 13 (𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8382rexlimivw 3058 . . . . . . . . . . . 12 (∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈) → (𝑎𝑏 → ¬ 𝑎 ran 𝑈))
8483impcom 445 . . . . . . . . . . 11 ((𝑎𝑏 ∧ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
8584exlimiv 1898 . . . . . . . . . 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 2651 . . . . . . . . . . 11 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
8887rnmpt 5403 . . . . . . . . . 10 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
8988unieqi 4477 . . . . . . . . 9 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = {𝑏 ∣ ∃𝑧𝑃 𝑏 = ((𝑡𝑧) ∖ ran 𝑈)}
9086, 89eleq2s 2748 . . . . . . . 8 (𝑎 ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) → ¬ 𝑎 ran 𝑈)
9178, 90mprgbir 2956 . . . . . . 7 ( ran (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∩ ran 𝑈) = ∅
92 ssdisj 4059 . . . . . . 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 708 . . . . . 6 ( ran ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) ∩ ran 𝑈) = ∅
9475, 93syl6eq 2701 . . . . 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 481 . . 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 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  𝒫 cpw 4191  ∪ cuni 4468  ∩ cint 4507   class class class wbr 4685   ↦ cmpt 4762  dom cdm 5143  ran crn 5144   ∘ ccom 5147  suc csuc 5763  Fun wfun 5920  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926  ℩crio 6650  (class class class)co 6690   ↦ cmpt2 6692  ωcom 7107  seq𝜔cseqom 7587   ↑𝑚 cmap 7899   ≈ cen 7994  Fincfn 7997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803 This theorem is referenced by:  fin23lem31  9203
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