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Theorem ptcnplem 23630
Description: Lemma for ptcnp 23631. (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 4237 . . . 4 (𝐼𝑊) ⊆ 𝑊
3 ssfi 9214 . . . 4 ((𝑊 ∈ Fin ∧ (𝐼𝑊) ⊆ 𝑊) → (𝐼𝑊) ∈ Fin)
41, 2, 3sylancl 586 . . 3 ((𝜑𝜓) → (𝐼𝑊) ∈ Fin)
5 nfv 1913 . . . . 5 𝑘𝜑
6 ptcnplem.1 . . . . 5 𝑘𝜓
75, 6nfan 1898 . . . 4 𝑘(𝜑𝜓)
8 elinel1 4200 . . . . . 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 7103 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
19 toptopon2 22925 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
2018, 19sylib 218 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
21 cnpf2 23259 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2216, 20, 9, 21syl3anc 1372 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
23 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2423fmpt 7129 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2522, 24sylibr 234 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
2625r19.21bi 3250 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
2723fvmpt2 7026 . . . . . . . . . . . . . . . . 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 5241 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼𝐴))
31 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → 𝑥𝑋)
32 ptcnp.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
3332adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝐼𝑉)
3433mptexd 7245 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ V)
35 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = (𝑥𝑋 ↦ (𝑘𝐼𝐴))
3635fvmpt2 7026 . . . . . . . . . . . . . . 15 ((𝑥𝑋 ∧ (𝑘𝐼𝐴) ∈ V) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
3731, 34, 36syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
3830, 37eqtr4d 2779 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
3938ralrimiva 3145 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
4039adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
41 nfcv 2904 . . . . . . . . . . . . . 14 𝑥𝐼
42 nffvmpt1 6916 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝐷)
4341, 42nfmpt 5248 . . . . . . . . . . . . 13 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷))
44 nffvmpt1 6916 . . . . . . . . . . . . 13 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
4543, 44nfeq 2918 . . . . . . . . . . . 12 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
46 fveq2 6905 . . . . . . . . . . . . . 14 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
4746mpteq2dv 5243 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)))
48 fveq2 6905 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷))
4947, 48eqeq12d 2752 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ↔ (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)))
5045, 49rspc 3609 . . . . . . . . . . 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 2840 . . . . . . . . 9 ((𝜑𝜓) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘))
5432adantr 480 . . . . . . . . . 10 ((𝜑𝜓) → 𝐼𝑉)
55 mptelixpg 8976 . . . . . . . . . 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 3250 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘))
59 cnpimaex 23265 . . . . . . 7 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷) ∧ (𝐺𝑘) ∈ (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
6010, 11, 58, 59syl3anc 1372 . . . . . 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 3256 . . 3 ((𝜑𝜓) → ∀𝑘 ∈ (𝐼𝑊)∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
64 eleq2 2829 . . . . 5 (𝑢 = (𝑓𝑘) → (𝐷𝑢𝐷 ∈ (𝑓𝑘)))
65 imaeq2 6073 . . . . . 6 (𝑢 = (𝑓𝑘) → ((𝑥𝑋𝐴) “ 𝑢) = ((𝑥𝑋𝐴) “ (𝑓𝑘)))
6665sseq1d 4014 . . . . 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 9321 . . 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 22921 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
7270, 71syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑋 = 𝐽)
7372ineq1d 4218 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) = ( 𝐽 ran 𝑓))
74 topontop 22920 . . . . . . 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 6742 . . . . . 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 6735 . . . . . . . 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 6825 . . . . . . 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 9352 . . . . . 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 2736 . . . . . 6 𝐽 = 𝐽
8786rintopn 22916 . . . . 5 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ran 𝑓 ∈ Fin) → ( 𝐽 ran 𝑓) ∈ 𝐽)
8876, 78, 85, 87syl3anc 1372 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ( 𝐽 ran 𝑓) ∈ 𝐽)
8973, 88eqeltrd 2840 . . 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 3082 . . . . . 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 2829 . . . . . . 7 (𝑧 = (𝑓𝑘) → (𝐷𝑧𝐷 ∈ (𝑓𝑘)))
9594ralrn 7107 . . . . . 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 4988 . . . 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 1913 . . . . . . . . . 10 𝑘 𝑓:(𝐼𝑊)⟶𝐽
1017, 100nfan 1898 . . . . . . . . 9 𝑘((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽)
102 funmpt 6603 . . . . . . . . . . . . 13 Fun (𝑥𝑋𝐴)
103 simp-4l 782 . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓:(𝐼𝑊)⟶𝐽)
106 simplr 768 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑘 ∈ (𝐼𝑊))
107105, 106ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ∈ 𝐽)
108 toponss 22934 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑓𝑘) ∈ 𝐽) → (𝑓𝑘) ⊆ 𝑋)
109104, 107, 108syl2anc 584 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ 𝑋)
110106elin1d 4203 . . . . . . . . . . . . . . . 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 6261 . . . . . . . . . . . . . . 15 (∀𝑥𝑋 𝐴 (𝐹𝑘) → dom (𝑥𝑋𝐴) = 𝑋)
113111, 112syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → dom (𝑥𝑋𝐴) = 𝑋)
114109, 113sseqtrrd 4020 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ dom (𝑥𝑋𝐴))
115 funimass4 6972 . . . . . . . . . . . . 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 6916 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝑡)
118117nfel1 2921 . . . . . . . . . . . . 13 𝑥((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)
119 nfv 1913 . . . . . . . . . . . . 13 𝑡((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)
120 fveq2 6905 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → ((𝑥𝑋𝐴)‘𝑡) = ((𝑥𝑋𝐴)‘𝑥))
121120eleq1d 2825 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘) ↔ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
122118, 119, 121cbvralw 3305 . . . . . . . . . . . 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 4236 . . . . . . . . . . . . 13 (𝑋 ran 𝑓) ⊆ 𝑋
125 ssralv 4051 . . . . . . . . . . . . 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 4237 . . . . . . . . . . . . . 14 (𝑋 ran 𝑓) ⊆ ran 𝑓
128105, 80syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓 Fn (𝐼𝑊))
129 fnfvelrn 7099 . . . . . . . . . . . . . . . 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 4962 . . . . . . . . . . . . . . 15 ((𝑓𝑘) ∈ ran 𝑓 ran 𝑓 ⊆ (𝑓𝑘))
132130, 131syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ran 𝑓 ⊆ (𝑓𝑘))
133127, 132sstrid 3994 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑋 ran 𝑓) ⊆ (𝑓𝑘))
134 ssralv 4051 . . . . . . . . . . . . 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 3110 . . . . . . . . . . . . 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 4200 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑋 ran 𝑓) → 𝑥𝑋)
138137, 27sylan 580 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
139138eleq1d 2825 . . . . . . . . . . . . . . . 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 3079 . . . . . . . . . . . . 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 3259 . . . . . . . 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 4130 . . . . . . . . . . . 12 (𝑘 ∈ (𝐼𝑊) → 𝑘𝐼)
151137, 26sylan2 593 . . . . . . . . . . . . 13 (((𝜑𝑘𝐼) ∧ 𝑥 ∈ (𝑋 ran 𝑓)) → 𝐴 (𝐹𝑘))
152151ralrimiva 3145 . . . . . . . . . . . 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 2829 . . . . . . . . . . . . 13 ((𝐺𝑘) = (𝐹𝑘) → (𝐴 ∈ (𝐺𝑘) ↔ 𝐴 (𝐹𝑘)))
156155ralbidv 3177 . . . . . . . . . . . 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 3256 . . . . . . . 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 4478 . . . . . . . . 9 ((𝐼𝑊) ∪ (𝐼𝑊)) = 𝐼
163162raleqi 3323 . . . . . . . 8 (∀𝑘 ∈ ((𝐼𝑊) ∪ (𝐼𝑊))∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
164 ralunb 4196 . . . . . . . 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 3288 . . . . . 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 6916 . . . . . . . . 9 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡)
171170nfel1 2921 . . . . . . . 8 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘)
172 nfv 1913 . . . . . . . 8 𝑡((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)
173 fveq2 6905 . . . . . . . . 9 (𝑡 = 𝑥 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
174173eleq1d 2825 . . . . . . . 8 (𝑡 = 𝑥 → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)))
175171, 172, 174cbvralw 3305 . . . . . . 7 (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘))
176 mptexg 7242 . . . . . . . . . . 11 (𝐼𝑉 → (𝑘𝐼𝐴) ∈ V)
177137, 176, 36syl2anr 597 . . . . . . . . . 10 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
178177eleq1d 2825 . . . . . . . . 9 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘)))
179 mptelixpg 8976 . . . . . . . . . 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 3175 . . . . . . 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 6603 . . . . 5 Fun (𝑥𝑋 ↦ (𝑘𝐼𝐴))
18732mptexd 7245 . . . . . . . . 9 (𝜑 → (𝑘𝐼𝐴) ∈ V)
188187ralrimivw 3149 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
189188ad2antrr 726 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
190 dmmptg 6261 . . . . . . 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 4026 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) ⊆ dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)))
193 funimass4 6972 . . . . 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 2829 . . . . 5 (𝑧 = (𝑋 ran 𝑓) → (𝐷𝑧𝐷 ∈ (𝑋 ran 𝑓)))
197 imaeq2 6073 . . . . . 6 (𝑧 = (𝑋 ran 𝑓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)))
198197sseq1d 4014 . . . . 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 3621 . . 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 836 . 2 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
20269, 201exlimddv 1934 1 ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wnf 1782  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950   cuni 4906   cint 4945  cmpt 5224  dom cdm 5684  ran crn 5685  cima 5687  Fun wfun 6554   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  Xcixp 8938  Fincfn 8986  tcpt 17484  Topctop 22900  TopOnctopon 22917   CnP ccnp 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-1o 8507  df-2o 8508  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-fin 8990  df-top 22901  df-topon 22918  df-cnp 23237
This theorem is referenced by:  ptcnp  23631
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