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Theorem ptcnplem 23645
Description: Lemma for ptcnp 23646. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
ptcnp.2 𝐾 = (∏t𝐹)
ptcnp.3 (𝜑𝐽 ∈ (TopOn‘𝑋))
ptcnp.4 (𝜑𝐼𝑉)
ptcnp.5 (𝜑𝐹:𝐼⟶Top)
ptcnp.6 (𝜑𝐷𝑋)
ptcnp.7 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
ptcnplem.1 𝑘𝜓
ptcnplem.2 ((𝜑𝜓) → 𝐺 Fn 𝐼)
ptcnplem.3 (((𝜑𝜓) ∧ 𝑘𝐼) → (𝐺𝑘) ∈ (𝐹𝑘))
ptcnplem.4 ((𝜑𝜓) → 𝑊 ∈ Fin)
ptcnplem.5 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝐺𝑘) = (𝐹𝑘))
ptcnplem.6 ((𝜑𝜓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝐺𝑘))
Assertion
Ref Expression
ptcnplem ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
Distinct variable groups:   𝑧,𝐴   𝑥,𝑘,𝑧,𝐷   𝑘,𝐼,𝑥,𝑧   𝑥,𝐺,𝑧   𝑘,𝐽,𝑧   𝑧,𝐾   𝜑,𝑘,𝑥,𝑧   𝑘,𝐹,𝑥,𝑧   𝑘,𝑉,𝑥   𝑘,𝑊,𝑧   𝑘,𝑋,𝑥,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑧,𝑘)   𝐴(𝑥,𝑘)   𝐺(𝑘)   𝐽(𝑥)   𝐾(𝑥,𝑘)   𝑉(𝑧)   𝑊(𝑥)

Proof of Theorem ptcnplem
Dummy variables 𝑓 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptcnplem.4 . . . 4 ((𝜑𝜓) → 𝑊 ∈ Fin)
2 inss2 4246 . . . 4 (𝐼𝑊) ⊆ 𝑊
3 ssfi 9212 . . . 4 ((𝑊 ∈ Fin ∧ (𝐼𝑊) ⊆ 𝑊) → (𝐼𝑊) ∈ Fin)
41, 2, 3sylancl 586 . . 3 ((𝜑𝜓) → (𝐼𝑊) ∈ Fin)
5 nfv 1912 . . . . 5 𝑘𝜑
6 ptcnplem.1 . . . . 5 𝑘𝜓
75, 6nfan 1897 . . . 4 𝑘(𝜑𝜓)
8 elinel1 4211 . . . . . 6 (𝑘 ∈ (𝐼𝑊) → 𝑘𝐼)
9 ptcnp.7 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
109adantlr 715 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
11 ptcnplem.3 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → (𝐺𝑘) ∈ (𝐹𝑘))
12 ptcnp.6 . . . . . . . . . . . 12 (𝜑𝐷𝑋)
1312adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → 𝐷𝑋)
14 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
15 ptcnp.3 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (TopOn‘𝑋))
1615adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
17 ptcnp.5 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:𝐼⟶Top)
1817ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
19 toptopon2 22940 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
2018, 19sylib 218 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
21 cnpf2 23274 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2216, 20, 9, 21syl3anc 1370 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
23 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2423fmpt 7130 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2522, 24sylibr 234 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
2625r19.21bi 3249 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
2723fvmpt2 7027 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝐴 (𝐹𝑘)) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2814, 26, 27syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2928an32s 652 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
3029mpteq2dva 5248 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼𝐴))
31 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → 𝑥𝑋)
32 ptcnp.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
3332adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝐼𝑉)
3433mptexd 7244 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ V)
35 eqid 2735 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = (𝑥𝑋 ↦ (𝑘𝐼𝐴))
3635fvmpt2 7027 . . . . . . . . . . . . . . 15 ((𝑥𝑋 ∧ (𝑘𝐼𝐴) ∈ V) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
3731, 34, 36syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
3830, 37eqtr4d 2778 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
3938ralrimiva 3144 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
4039adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
41 nfcv 2903 . . . . . . . . . . . . . 14 𝑥𝐼
42 nffvmpt1 6918 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝐷)
4341, 42nfmpt 5255 . . . . . . . . . . . . 13 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷))
44 nffvmpt1 6918 . . . . . . . . . . . . 13 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
4543, 44nfeq 2917 . . . . . . . . . . . 12 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
46 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
4746mpteq2dv 5250 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)))
48 fveq2 6907 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷))
4947, 48eqeq12d 2751 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ↔ (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)))
5045, 49rspc 3610 . . . . . . . . . . 11 (𝐷𝑋 → (∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)))
5113, 40, 50sylc 65 . . . . . . . . . 10 ((𝜑𝜓) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷))
52 ptcnplem.6 . . . . . . . . . 10 ((𝜑𝜓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝐺𝑘))
5351, 52eqeltrd 2839 . . . . . . . . 9 ((𝜑𝜓) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘))
5432adantr 480 . . . . . . . . . 10 ((𝜑𝜓) → 𝐼𝑉)
55 mptelixpg 8974 . . . . . . . . . 10 (𝐼𝑉 → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)))
5654, 55syl 17 . . . . . . . . 9 ((𝜑𝜓) → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)))
5753, 56mpbid 232 . . . . . . . 8 ((𝜑𝜓) → ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘))
5857r19.21bi 3249 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘))
59 cnpimaex 23280 . . . . . . 7 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷) ∧ (𝐺𝑘) ∈ (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
6010, 11, 58, 59syl3anc 1370 . . . . . 6 (((𝜑𝜓) ∧ 𝑘𝐼) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
618, 60sylan2 593 . . . . 5 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
6261ex 412 . . . 4 ((𝜑𝜓) → (𝑘 ∈ (𝐼𝑊) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘))))
637, 62ralrimi 3255 . . 3 ((𝜑𝜓) → ∀𝑘 ∈ (𝐼𝑊)∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
64 eleq2 2828 . . . . 5 (𝑢 = (𝑓𝑘) → (𝐷𝑢𝐷 ∈ (𝑓𝑘)))
65 imaeq2 6076 . . . . . 6 (𝑢 = (𝑓𝑘) → ((𝑥𝑋𝐴) “ 𝑢) = ((𝑥𝑋𝐴) “ (𝑓𝑘)))
6665sseq1d 4027 . . . . 5 (𝑢 = (𝑓𝑘) → (((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘) ↔ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))
6764, 66anbi12d 632 . . . 4 (𝑢 = (𝑓𝑘) → ((𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)) ↔ (𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
6867ac6sfi 9318 . . 3 (((𝐼𝑊) ∈ Fin ∧ ∀𝑘 ∈ (𝐼𝑊)∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘))) → ∃𝑓(𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
694, 63, 68syl2anc 584 . 2 ((𝜑𝜓) → ∃𝑓(𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
7015ad2antrr 726 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐽 ∈ (TopOn‘𝑋))
71 toponuni 22936 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
7270, 71syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑋 = 𝐽)
7372ineq1d 4227 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) = ( 𝐽 ran 𝑓))
74 topontop 22935 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
7515, 74syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
7675ad2antrr 726 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐽 ∈ Top)
77 frn 6744 . . . . . 6 (𝑓:(𝐼𝑊)⟶𝐽 → ran 𝑓𝐽)
7877ad2antrl 728 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ran 𝑓𝐽)
794adantr 480 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝐼𝑊) ∈ Fin)
80 ffn 6737 . . . . . . . 8 (𝑓:(𝐼𝑊)⟶𝐽𝑓 Fn (𝐼𝑊))
8180ad2antrl 728 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑓 Fn (𝐼𝑊))
82 dffn4 6827 . . . . . . 7 (𝑓 Fn (𝐼𝑊) ↔ 𝑓:(𝐼𝑊)–onto→ran 𝑓)
8381, 82sylib 218 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑓:(𝐼𝑊)–onto→ran 𝑓)
84 fofi 9349 . . . . . 6 (((𝐼𝑊) ∈ Fin ∧ 𝑓:(𝐼𝑊)–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
8579, 83, 84syl2anc 584 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ran 𝑓 ∈ Fin)
86 eqid 2735 . . . . . 6 𝐽 = 𝐽
8786rintopn 22931 . . . . 5 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ran 𝑓 ∈ Fin) → ( 𝐽 ran 𝑓) ∈ 𝐽)
8876, 78, 85, 87syl3anc 1370 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ( 𝐽 ran 𝑓) ∈ 𝐽)
8973, 88eqeltrd 2839 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) ∈ 𝐽)
9012ad2antrr 726 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐷𝑋)
91 simpl 482 . . . . . . 7 ((𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → 𝐷 ∈ (𝑓𝑘))
9291ralimi 3081 . . . . . 6 (∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘))
9392ad2antll 729 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘))
94 eleq2 2828 . . . . . . 7 (𝑧 = (𝑓𝑘) → (𝐷𝑧𝐷 ∈ (𝑓𝑘)))
9594ralrn 7108 . . . . . 6 (𝑓 Fn (𝐼𝑊) → (∀𝑧 ∈ ran 𝑓 𝐷𝑧 ↔ ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘)))
9681, 95syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (∀𝑧 ∈ ran 𝑓 𝐷𝑧 ↔ ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘)))
9793, 96mpbird 257 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑧 ∈ ran 𝑓 𝐷𝑧)
98 elrint 4994 . . . 4 (𝐷 ∈ (𝑋 ran 𝑓) ↔ (𝐷𝑋 ∧ ∀𝑧 ∈ ran 𝑓 𝐷𝑧))
9990, 97, 98sylanbrc 583 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐷 ∈ (𝑋 ran 𝑓))
100 nfv 1912 . . . . . . . . . 10 𝑘 𝑓:(𝐼𝑊)⟶𝐽
1017, 100nfan 1897 . . . . . . . . 9 𝑘((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽)
102 funmpt 6606 . . . . . . . . . . . . 13 Fun (𝑥𝑋𝐴)
103 simp-4l 783 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝜑)
104103, 15syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝐽 ∈ (TopOn‘𝑋))
105 simpllr 776 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓:(𝐼𝑊)⟶𝐽)
106 simplr 769 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑘 ∈ (𝐼𝑊))
107105, 106ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ∈ 𝐽)
108 toponss 22949 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑓𝑘) ∈ 𝐽) → (𝑓𝑘) ⊆ 𝑋)
109104, 107, 108syl2anc 584 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ 𝑋)
110106elin1d 4214 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑘𝐼)
111103, 110, 25syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
112 dmmptg 6264 . . . . . . . . . . . . . . 15 (∀𝑥𝑋 𝐴 (𝐹𝑘) → dom (𝑥𝑋𝐴) = 𝑋)
113111, 112syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → dom (𝑥𝑋𝐴) = 𝑋)
114109, 113sseqtrrd 4037 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ dom (𝑥𝑋𝐴))
115 funimass4 6973 . . . . . . . . . . . . 13 ((Fun (𝑥𝑋𝐴) ∧ (𝑓𝑘) ⊆ dom (𝑥𝑋𝐴)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)))
116102, 114, 115sylancr 587 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)))
117 nffvmpt1 6918 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝑡)
118117nfel1 2920 . . . . . . . . . . . . 13 𝑥((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)
119 nfv 1912 . . . . . . . . . . . . 13 𝑡((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)
120 fveq2 6907 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → ((𝑥𝑋𝐴)‘𝑡) = ((𝑥𝑋𝐴)‘𝑥))
121120eleq1d 2824 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘) ↔ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
122118, 119, 121cbvralw 3304 . . . . . . . . . . . 12 (∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘))
123116, 122bitrdi 287 . . . . . . . . . . 11 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
124 inss1 4245 . . . . . . . . . . . . 13 (𝑋 ran 𝑓) ⊆ 𝑋
125 ssralv 4064 . . . . . . . . . . . . 13 ((𝑋 ran 𝑓) ⊆ 𝑋 → (∀𝑥𝑋 𝐴 (𝐹𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
126124, 111, 125mpsyl 68 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
127 inss2 4246 . . . . . . . . . . . . . 14 (𝑋 ran 𝑓) ⊆ ran 𝑓
128105, 80syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓 Fn (𝐼𝑊))
129 fnfvelrn 7100 . . . . . . . . . . . . . . . 16 ((𝑓 Fn (𝐼𝑊) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝑓𝑘) ∈ ran 𝑓)
130128, 106, 129syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ∈ ran 𝑓)
131 intss1 4968 . . . . . . . . . . . . . . 15 ((𝑓𝑘) ∈ ran 𝑓 ran 𝑓 ⊆ (𝑓𝑘))
132130, 131syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ran 𝑓 ⊆ (𝑓𝑘))
133127, 132sstrid 4007 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑋 ran 𝑓) ⊆ (𝑓𝑘))
134 ssralv 4064 . . . . . . . . . . . . 13 ((𝑋 ran 𝑓) ⊆ (𝑓𝑘) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
135133, 134syl 17 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
136 r19.26 3109 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋 ran 𝑓)(𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) ↔ (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘) ∧ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
137 elinel1 4211 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑋 ran 𝑓) → 𝑥𝑋)
138137, 27sylan 580 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
139138eleq1d 2824 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → (((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) ↔ 𝐴 ∈ (𝐺𝑘)))
140139biimpd 229 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → (((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → 𝐴 ∈ (𝐺𝑘)))
141140expimpd 453 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋 ran 𝑓) → ((𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → 𝐴 ∈ (𝐺𝑘)))
142141ralimia 3078 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋 ran 𝑓)(𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
143136, 142sylbir 235 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘) ∧ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
144126, 135, 143syl6an 684 . . . . . . . . . . 11 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
145123, 144sylbid 240 . . . . . . . . . 10 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
146145expimpd 453 . . . . . . . . 9 ((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) → ((𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
147101, 146ralimdaa 3258 . . . . . . . 8 (((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) → (∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
148147impr 454 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
149 simpl 482 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝜑)
150 eldifi 4141 . . . . . . . . . . . 12 (𝑘 ∈ (𝐼𝑊) → 𝑘𝐼)
151137, 26sylan2 593 . . . . . . . . . . . . 13 (((𝜑𝑘𝐼) ∧ 𝑥 ∈ (𝑋 ran 𝑓)) → 𝐴 (𝐹𝑘))
152151ralrimiva 3144 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
153149, 150, 152syl2an 596 . . . . . . . . . . 11 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
154 ptcnplem.5 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝐺𝑘) = (𝐹𝑘))
155 eleq2 2828 . . . . . . . . . . . . 13 ((𝐺𝑘) = (𝐹𝑘) → (𝐴 ∈ (𝐺𝑘) ↔ 𝐴 (𝐹𝑘)))
156155ralbidv 3176 . . . . . . . . . . . 12 ((𝐺𝑘) = (𝐹𝑘) → (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
157154, 156syl 17 . . . . . . . . . . 11 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
158153, 157mpbird 257 . . . . . . . . . 10 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
159158ex 412 . . . . . . . . 9 ((𝜑𝜓) → (𝑘 ∈ (𝐼𝑊) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
1607, 159ralrimi 3255 . . . . . . . 8 ((𝜑𝜓) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
161160adantr 480 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
162 inundif 4485 . . . . . . . . 9 ((𝐼𝑊) ∪ (𝐼𝑊)) = 𝐼
163162raleqi 3322 . . . . . . . 8 (∀𝑘 ∈ ((𝐼𝑊) ∪ (𝐼𝑊))∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
164 ralunb 4207 . . . . . . . 8 (∀𝑘 ∈ ((𝐼𝑊) ∪ (𝐼𝑊))∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ (∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ∧ ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
165163, 164bitr3i 277 . . . . . . 7 (∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ (∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ∧ ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
166148, 161, 165sylanbrc 583 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
167 ralcom 3287 . . . . . 6 (∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘) ↔ ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
168166, 167sylibr 234 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘))
16932ad2antrr 726 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐼𝑉)
170 nffvmpt1 6918 . . . . . . . . 9 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡)
171170nfel1 2920 . . . . . . . 8 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘)
172 nfv 1912 . . . . . . . 8 𝑡((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)
173 fveq2 6907 . . . . . . . . 9 (𝑡 = 𝑥 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
174173eleq1d 2824 . . . . . . . 8 (𝑡 = 𝑥 → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)))
175171, 172, 174cbvralw 3304 . . . . . . 7 (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘))
176 mptexg 7241 . . . . . . . . . . 11 (𝐼𝑉 → (𝑘𝐼𝐴) ∈ V)
177137, 176, 36syl2anr 597 . . . . . . . . . 10 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
178177eleq1d 2824 . . . . . . . . 9 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘)))
179 mptelixpg 8974 . . . . . . . . . 10 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
180179adantr 480 . . . . . . . . 9 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
181178, 180bitrd 279 . . . . . . . 8 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
182181ralbidva 3174 . . . . . . 7 (𝐼𝑉 → (∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
183175, 182bitrid 283 . . . . . 6 (𝐼𝑉 → (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
184169, 183syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
185168, 184mpbird 257 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘))
186 funmpt 6606 . . . . 5 Fun (𝑥𝑋 ↦ (𝑘𝐼𝐴))
18732mptexd 7244 . . . . . . . . 9 (𝜑 → (𝑘𝐼𝐴) ∈ V)
188187ralrimivw 3148 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
189188ad2antrr 726 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
190 dmmptg 6264 . . . . . . 7 (∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V → dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = 𝑋)
191189, 190syl 17 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = 𝑋)
192124, 191sseqtrrid 4049 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) ⊆ dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)))
193 funimass4 6973 . . . . 5 ((Fun (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∧ (𝑋 ran 𝑓) ⊆ dom (𝑥𝑋 ↦ (𝑘𝐼𝐴))) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘)))
194186, 192, 193sylancr 587 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘)))
195185, 194mpbird 257 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))
196 eleq2 2828 . . . . 5 (𝑧 = (𝑋 ran 𝑓) → (𝐷𝑧𝐷 ∈ (𝑋 ran 𝑓)))
197 imaeq2 6076 . . . . . 6 (𝑧 = (𝑋 ran 𝑓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)))
198197sseq1d 4027 . . . . 5 (𝑧 = (𝑋 ran 𝑓) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘)))
199196, 198anbi12d 632 . . . 4 (𝑧 = (𝑋 ran 𝑓) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)) ↔ (𝐷 ∈ (𝑋 ran 𝑓) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))))
200199rspcev 3622 . . 3 (((𝑋 ran 𝑓) ∈ 𝐽 ∧ (𝐷 ∈ (𝑋 ran 𝑓) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
20189, 99, 195, 200syl12anc 837 . 2 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
20269, 201exlimddv 1933 1 ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wnf 1780  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963   cuni 4912   cint 4951  cmpt 5231  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  Xcixp 8936  Fincfn 8984  tcpt 17485  Topctop 22915  TopOnctopon 22932   CnP ccnp 23249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-top 22916  df-topon 22933  df-cnp 23252
This theorem is referenced by:  ptcnp  23646
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