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Theorem ptcnplem 21472
Description: Lemma for ptcnp 21473. (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 3867 . . . 4 (𝐼𝑊) ⊆ 𝑊
3 ssfi 8221 . . . 4 ((𝑊 ∈ Fin ∧ (𝐼𝑊) ⊆ 𝑊) → (𝐼𝑊) ∈ Fin)
41, 2, 3sylancl 695 . . 3 ((𝜑𝜓) → (𝐼𝑊) ∈ Fin)
5 nfv 1883 . . . . 5 𝑘𝜑
6 ptcnplem.1 . . . . 5 𝑘𝜓
75, 6nfan 1868 . . . 4 𝑘(𝜑𝜓)
8 inss1 3866 . . . . . . 7 (𝐼𝑊) ⊆ 𝐼
98sseli 3632 . . . . . 6 (𝑘 ∈ (𝐼𝑊) → 𝑘𝐼)
10 ptcnp.7 . . . . . . . 8 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
1110adantlr 751 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))
12 ptcnplem.3 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → (𝐺𝑘) ∈ (𝐹𝑘))
13 ptcnp.6 . . . . . . . . . . . 12 (𝜑𝐷𝑋)
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → 𝐷𝑋)
15 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
16 ptcnp.3 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (TopOn‘𝑋))
1716adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐼) → 𝐽 ∈ (TopOn‘𝑋))
18 ptcnp.5 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐹:𝐼⟶Top)
1918ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ Top)
20 eqid 2651 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹𝑘) = (𝐹𝑘)
2120toptopon 20770 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑘) ∈ Top ↔ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
2219, 21sylib 208 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝐼) → (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)))
23 cnpf2 21102 . . . . . . . . . . . . . . . . . . . 20 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹𝑘) ∈ (TopOn‘ (𝐹𝑘)) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷)) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2417, 22, 10, 23syl3anc 1366 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝐼) → (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
25 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2625fmpt 6421 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑋 𝐴 (𝐹𝑘) ↔ (𝑥𝑋𝐴):𝑋 (𝐹𝑘))
2724, 26sylibr 224 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
2827r19.21bi 2961 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → 𝐴 (𝐹𝑘))
2925fvmpt2 6330 . . . . . . . . . . . . . . . . 17 ((𝑥𝑋𝐴 (𝐹𝑘)) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
3015, 28, 29syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
3130an32s 863 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑋) ∧ 𝑘𝐼) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
3231mpteq2dva 4777 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼𝐴))
33 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → 𝑥𝑋)
34 ptcnp.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐼𝑉)
3534adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝐼𝑉)
36 mptexg 6525 . . . . . . . . . . . . . . . 16 (𝐼𝑉 → (𝑘𝐼𝐴) ∈ V)
3735, 36syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (𝑘𝐼𝐴) ∈ V)
38 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = (𝑥𝑋 ↦ (𝑘𝐼𝐴))
3938fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑥𝑋 ∧ (𝑘𝐼𝐴) ∈ V) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
4033, 37, 39syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
4132, 40eqtr4d 2688 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
4241ralrimiva 2995 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
4342adantr 480 . . . . . . . . . . 11 ((𝜑𝜓) → ∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
44 nfcv 2793 . . . . . . . . . . . . . 14 𝑥𝐼
45 nffvmpt1 6237 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝐷)
4644, 45nfmpt 4779 . . . . . . . . . . . . 13 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷))
47 nffvmpt1 6237 . . . . . . . . . . . . 13 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
4846, 47nfeq 2805 . . . . . . . . . . . 12 𝑥(𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)
49 fveq2 6229 . . . . . . . . . . . . . 14 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
5049mpteq2dv 4778 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)))
51 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷))
5250, 51eqeq12d 2666 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ↔ (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)))
5348, 52rspc 3334 . . . . . . . . . . 11 (𝐷𝑋 → (∀𝑥𝑋 (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝑥)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷)))
5414, 43, 53sylc 65 . . . . . . . . . 10 ((𝜑𝜓) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷))
55 ptcnplem.6 . . . . . . . . . 10 ((𝜑𝜓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝐺𝑘))
5654, 55eqeltrd 2730 . . . . . . . . 9 ((𝜑𝜓) → (𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘))
5734adantr 480 . . . . . . . . . 10 ((𝜑𝜓) → 𝐼𝑉)
58 mptelixpg 7987 . . . . . . . . . 10 (𝐼𝑉 → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)))
5957, 58syl 17 . . . . . . . . 9 ((𝜑𝜓) → ((𝑘𝐼 ↦ ((𝑥𝑋𝐴)‘𝐷)) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)))
6056, 59mpbid 222 . . . . . . . 8 ((𝜑𝜓) → ∀𝑘𝐼 ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘))
6160r19.21bi 2961 . . . . . . 7 (((𝜑𝜓) ∧ 𝑘𝐼) → ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘))
62 cnpimaex 21108 . . . . . . 7 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷) ∧ (𝐺𝑘) ∈ (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ (𝐺𝑘)) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
6311, 12, 61, 62syl3anc 1366 . . . . . 6 (((𝜑𝜓) ∧ 𝑘𝐼) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
649, 63sylan2 490 . . . . 5 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
6564ex 449 . . . 4 ((𝜑𝜓) → (𝑘 ∈ (𝐼𝑊) → ∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘))))
667, 65ralrimi 2986 . . 3 ((𝜑𝜓) → ∀𝑘 ∈ (𝐼𝑊)∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)))
67 eleq2 2719 . . . . 5 (𝑢 = (𝑓𝑘) → (𝐷𝑢𝐷 ∈ (𝑓𝑘)))
68 imaeq2 5497 . . . . . 6 (𝑢 = (𝑓𝑘) → ((𝑥𝑋𝐴) “ 𝑢) = ((𝑥𝑋𝐴) “ (𝑓𝑘)))
6968sseq1d 3665 . . . . 5 (𝑢 = (𝑓𝑘) → (((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘) ↔ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))
7067, 69anbi12d 747 . . . 4 (𝑢 = (𝑓𝑘) → ((𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘)) ↔ (𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
7170ac6sfi 8245 . . 3 (((𝐼𝑊) ∈ Fin ∧ ∀𝑘 ∈ (𝐼𝑊)∃𝑢𝐽 (𝐷𝑢 ∧ ((𝑥𝑋𝐴) “ 𝑢) ⊆ (𝐺𝑘))) → ∃𝑓(𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
724, 66, 71syl2anc 694 . 2 ((𝜑𝜓) → ∃𝑓(𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘))))
7316ad2antrr 762 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐽 ∈ (TopOn‘𝑋))
74 toponuni 20767 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
7573, 74syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑋 = 𝐽)
7675ineq1d 3846 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) = ( 𝐽 ran 𝑓))
77 topontop 20766 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
7816, 77syl 17 . . . . . 6 (𝜑𝐽 ∈ Top)
7978ad2antrr 762 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐽 ∈ Top)
80 frn 6091 . . . . . 6 (𝑓:(𝐼𝑊)⟶𝐽 → ran 𝑓𝐽)
8180ad2antrl 764 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ran 𝑓𝐽)
824adantr 480 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝐼𝑊) ∈ Fin)
83 ffn 6083 . . . . . . . 8 (𝑓:(𝐼𝑊)⟶𝐽𝑓 Fn (𝐼𝑊))
8483ad2antrl 764 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑓 Fn (𝐼𝑊))
85 dffn4 6159 . . . . . . 7 (𝑓 Fn (𝐼𝑊) ↔ 𝑓:(𝐼𝑊)–onto→ran 𝑓)
8684, 85sylib 208 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝑓:(𝐼𝑊)–onto→ran 𝑓)
87 fofi 8293 . . . . . 6 (((𝐼𝑊) ∈ Fin ∧ 𝑓:(𝐼𝑊)–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
8882, 86, 87syl2anc 694 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ran 𝑓 ∈ Fin)
89 eqid 2651 . . . . . 6 𝐽 = 𝐽
9089rintopn 20762 . . . . 5 ((𝐽 ∈ Top ∧ ran 𝑓𝐽 ∧ ran 𝑓 ∈ Fin) → ( 𝐽 ran 𝑓) ∈ 𝐽)
9179, 81, 88, 90syl3anc 1366 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ( 𝐽 ran 𝑓) ∈ 𝐽)
9276, 91eqeltrd 2730 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) ∈ 𝐽)
9313ad2antrr 762 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐷𝑋)
94 simpl 472 . . . . . . 7 ((𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → 𝐷 ∈ (𝑓𝑘))
9594ralimi 2981 . . . . . 6 (∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘))
9695ad2antll 765 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘))
97 eleq2 2719 . . . . . . 7 (𝑧 = (𝑓𝑘) → (𝐷𝑧𝐷 ∈ (𝑓𝑘)))
9897ralrn 6402 . . . . . 6 (𝑓 Fn (𝐼𝑊) → (∀𝑧 ∈ ran 𝑓 𝐷𝑧 ↔ ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘)))
9984, 98syl 17 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (∀𝑧 ∈ ran 𝑓 𝐷𝑧 ↔ ∀𝑘 ∈ (𝐼𝑊)𝐷 ∈ (𝑓𝑘)))
10096, 99mpbird 247 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑧 ∈ ran 𝑓 𝐷𝑧)
101 elrint 4550 . . . 4 (𝐷 ∈ (𝑋 ran 𝑓) ↔ (𝐷𝑋 ∧ ∀𝑧 ∈ ran 𝑓 𝐷𝑧))
10293, 100, 101sylanbrc 699 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐷 ∈ (𝑋 ran 𝑓))
103 nfv 1883 . . . . . . . . . 10 𝑘 𝑓:(𝐼𝑊)⟶𝐽
1047, 103nfan 1868 . . . . . . . . 9 𝑘((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽)
105 funmpt 5964 . . . . . . . . . . . . 13 Fun (𝑥𝑋𝐴)
106 simp-4l 823 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝜑)
107106, 16syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝐽 ∈ (TopOn‘𝑋))
108 simpllr 815 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓:(𝐼𝑊)⟶𝐽)
109 simplr 807 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑘 ∈ (𝐼𝑊))
110108, 109ffvelrnd 6400 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ∈ 𝐽)
111 toponss 20779 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑓𝑘) ∈ 𝐽) → (𝑓𝑘) ⊆ 𝑋)
112107, 110, 111syl2anc 694 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ 𝑋)
1138, 109sseldi 3634 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑘𝐼)
114106, 113, 27syl2anc 694 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ∀𝑥𝑋 𝐴 (𝐹𝑘))
115 dmmptg 5670 . . . . . . . . . . . . . . 15 (∀𝑥𝑋 𝐴 (𝐹𝑘) → dom (𝑥𝑋𝐴) = 𝑋)
116114, 115syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → dom (𝑥𝑋𝐴) = 𝑋)
117112, 116sseqtr4d 3675 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ⊆ dom (𝑥𝑋𝐴))
118 funimass4 6286 . . . . . . . . . . . . 13 ((Fun (𝑥𝑋𝐴) ∧ (𝑓𝑘) ⊆ dom (𝑥𝑋𝐴)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)))
119105, 117, 118sylancr 696 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)))
120 nffvmpt1 6237 . . . . . . . . . . . . . 14 𝑥((𝑥𝑋𝐴)‘𝑡)
121120nfel1 2808 . . . . . . . . . . . . 13 𝑥((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘)
122 nfv 1883 . . . . . . . . . . . . 13 𝑡((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)
123 fveq2 6229 . . . . . . . . . . . . . 14 (𝑡 = 𝑥 → ((𝑥𝑋𝐴)‘𝑡) = ((𝑥𝑋𝐴)‘𝑥))
124123eleq1d 2715 . . . . . . . . . . . . 13 (𝑡 = 𝑥 → (((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘) ↔ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
125121, 122, 124cbvral 3197 . . . . . . . . . . . 12 (∀𝑡 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑡) ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘))
126119, 125syl6bb 276 . . . . . . . . . . 11 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
127 inss1 3866 . . . . . . . . . . . . 13 (𝑋 ran 𝑓) ⊆ 𝑋
128 ssralv 3699 . . . . . . . . . . . . 13 ((𝑋 ran 𝑓) ⊆ 𝑋 → (∀𝑥𝑋 𝐴 (𝐹𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
129127, 114, 128mpsyl 68 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
130 inss2 3867 . . . . . . . . . . . . . 14 (𝑋 ran 𝑓) ⊆ ran 𝑓
131108, 83syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → 𝑓 Fn (𝐼𝑊))
132 fnfvelrn 6396 . . . . . . . . . . . . . . . 16 ((𝑓 Fn (𝐼𝑊) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝑓𝑘) ∈ ran 𝑓)
133131, 109, 132syl2anc 694 . . . . . . . . . . . . . . 15 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑓𝑘) ∈ ran 𝑓)
134 intss1 4524 . . . . . . . . . . . . . . 15 ((𝑓𝑘) ∈ ran 𝑓 ran 𝑓 ⊆ (𝑓𝑘))
135133, 134syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → ran 𝑓 ⊆ (𝑓𝑘))
136130, 135syl5ss 3647 . . . . . . . . . . . . 13 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (𝑋 ran 𝑓) ⊆ (𝑓𝑘))
137 ssralv 3699 . . . . . . . . . . . . 13 ((𝑋 ran 𝑓) ⊆ (𝑓𝑘) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
138136, 137syl 17 . . . . . . . . . . . 12 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
139 r19.26 3093 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋 ran 𝑓)(𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) ↔ (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘) ∧ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)))
140127sseli 3632 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝑋 ran 𝑓) → 𝑥𝑋)
141140, 29sylan 487 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
142141eleq1d 2715 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → (((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) ↔ 𝐴 ∈ (𝐺𝑘)))
143142biimpd 219 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝑋 ran 𝑓) ∧ 𝐴 (𝐹𝑘)) → (((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → 𝐴 ∈ (𝐺𝑘)))
144143expimpd 628 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋 ran 𝑓) → ((𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → 𝐴 ∈ (𝐺𝑘)))
145144ralimia 2979 . . . . . . . . . . . . 13 (∀𝑥 ∈ (𝑋 ran 𝑓)(𝐴 (𝐹𝑘) ∧ ((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
146139, 145sylbir 225 . . . . . . . . . . . 12 ((∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘) ∧ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
147129, 138, 146syl6an 567 . . . . . . . . . . 11 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (∀𝑥 ∈ (𝑓𝑘)((𝑥𝑋𝐴)‘𝑥) ∈ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
148126, 147sylbid 230 . . . . . . . . . 10 (((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) ∧ 𝐷 ∈ (𝑓𝑘)) → (((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
149148expimpd 628 . . . . . . . . 9 ((((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) ∧ 𝑘 ∈ (𝐼𝑊)) → ((𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
150104, 149ralimdaa 2987 . . . . . . . 8 (((𝜑𝜓) ∧ 𝑓:(𝐼𝑊)⟶𝐽) → (∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
151150impr 648 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
152 simpl 472 . . . . . . . . . . . 12 ((𝜑𝜓) → 𝜑)
153 eldifi 3765 . . . . . . . . . . . 12 (𝑘 ∈ (𝐼𝑊) → 𝑘𝐼)
154140, 28sylan2 490 . . . . . . . . . . . . 13 (((𝜑𝑘𝐼) ∧ 𝑥 ∈ (𝑋 ran 𝑓)) → 𝐴 (𝐹𝑘))
155154ralrimiva 2995 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
156152, 153, 155syl2an 493 . . . . . . . . . . 11 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘))
157 ptcnplem.5 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝐺𝑘) = (𝐹𝑘))
158 eleq2 2719 . . . . . . . . . . . . 13 ((𝐺𝑘) = (𝐹𝑘) → (𝐴 ∈ (𝐺𝑘) ↔ 𝐴 (𝐹𝑘)))
159158ralbidv 3015 . . . . . . . . . . . 12 ((𝐺𝑘) = (𝐹𝑘) → (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
160157, 159syl 17 . . . . . . . . . . 11 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 (𝐹𝑘)))
161156, 160mpbird 247 . . . . . . . . . 10 (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
162161ex 449 . . . . . . . . 9 ((𝜑𝜓) → (𝑘 ∈ (𝐼𝑊) → ∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
1637, 162ralrimi 2986 . . . . . . . 8 ((𝜑𝜓) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
164163adantr 480 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
165 inundif 4079 . . . . . . . . 9 ((𝐼𝑊) ∪ (𝐼𝑊)) = 𝐼
166165raleqi 3172 . . . . . . . 8 (∀𝑘 ∈ ((𝐼𝑊) ∪ (𝐼𝑊))∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
167 ralunb 3827 . . . . . . . 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)))
168166, 167bitr3i 266 . . . . . . 7 (∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ↔ (∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘) ∧ ∀𝑘 ∈ (𝐼𝑊)∀𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘)))
169151, 164, 168sylanbrc 699 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
170 ralcom 3127 . . . . . 6 (∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘) ↔ ∀𝑘𝐼𝑥 ∈ (𝑋 ran 𝑓)𝐴 ∈ (𝐺𝑘))
171169, 170sylibr 224 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘))
17234ad2antrr 762 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → 𝐼𝑉)
173 nffvmpt1 6237 . . . . . . . . 9 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡)
174173nfel1 2808 . . . . . . . 8 𝑥((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘)
175 nfv 1883 . . . . . . . 8 𝑡((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)
176 fveq2 6229 . . . . . . . . 9 (𝑡 = 𝑥 → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥))
177176eleq1d 2715 . . . . . . . 8 (𝑡 = 𝑥 → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘)))
178174, 175, 177cbvral 3197 . . . . . . 7 (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘))
179140, 36, 39syl2anr 494 . . . . . . . . . 10 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) = (𝑘𝐼𝐴))
180179eleq1d 2715 . . . . . . . . 9 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ (𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘)))
181 mptelixpg 7987 . . . . . . . . . 10 (𝐼𝑉 → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
182181adantr 480 . . . . . . . . 9 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → ((𝑘𝐼𝐴) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
183180, 182bitrd 268 . . . . . . . 8 ((𝐼𝑉𝑥 ∈ (𝑋 ran 𝑓)) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
184183ralbidva 3014 . . . . . . 7 (𝐼𝑉 → (∀𝑥 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑥) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
185178, 184syl5bb 272 . . . . . 6 (𝐼𝑉 → (∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘) ↔ ∀𝑥 ∈ (𝑋 ran 𝑓)∀𝑘𝐼 𝐴 ∈ (𝐺𝑘)))
186172, 185syl 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)))
187171, 186mpbird 247 . . . 4 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑡 ∈ (𝑋 ran 𝑓)((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝑡) ∈ X𝑘𝐼 (𝐺𝑘))
188 funmpt 5964 . . . . 5 Fun (𝑥𝑋 ↦ (𝑘𝐼𝐴))
18934, 36syl 17 . . . . . . . . 9 (𝜑 → (𝑘𝐼𝐴) ∈ V)
190189ralrimivw 2996 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
191190ad2antrr 762 . . . . . . 7 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V)
192 dmmptg 5670 . . . . . . 7 (∀𝑥𝑋 (𝑘𝐼𝐴) ∈ V → dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = 𝑋)
193191, 192syl 17 . . . . . 6 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)) = 𝑋)
194127, 193syl5sseqr 3687 . . . . 5 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → (𝑋 ran 𝑓) ⊆ dom (𝑥𝑋 ↦ (𝑘𝐼𝐴)))
195 funimass4 6286 . . . . 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)))
196188, 194, 195sylancr 696 . . . 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)))
197187, 196mpbird 247 . . 3 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))
198 eleq2 2719 . . . . 5 (𝑧 = (𝑋 ran 𝑓) → (𝐷𝑧𝐷 ∈ (𝑋 ran 𝑓)))
199 imaeq2 5497 . . . . . 6 (𝑧 = (𝑋 ran 𝑓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) = ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)))
200199sseq1d 3665 . . . . 5 (𝑧 = (𝑋 ran 𝑓) → (((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘) ↔ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘)))
201198, 200anbi12d 747 . . . 4 (𝑧 = (𝑋 ran 𝑓) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)) ↔ (𝐷 ∈ (𝑋 ran 𝑓) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))))
202201rspcev 3340 . . 3 (((𝑋 ran 𝑓) ∈ 𝐽 ∧ (𝐷 ∈ (𝑋 ran 𝑓) ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ (𝑋 ran 𝑓)) ⊆ X𝑘𝐼 (𝐺𝑘))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
20392, 102, 197, 202syl12anc 1364 . 2 (((𝜑𝜓) ∧ (𝑓:(𝐼𝑊)⟶𝐽 ∧ ∀𝑘 ∈ (𝐼𝑊)(𝐷 ∈ (𝑓𝑘) ∧ ((𝑥𝑋𝐴) “ (𝑓𝑘)) ⊆ (𝐺𝑘)))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
20472, 203exlimddv 1903 1 ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607   cuni 4468   cint 4507  cmpt 4762  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Xcixp 7950  Fincfn 7997  tcpt 16146  Topctop 20746  TopOnctopon 20763   CnP ccnp 21077
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-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-iin 4555  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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-fin 8001  df-top 20747  df-topon 20764  df-cnp 21080
This theorem is referenced by:  ptcnp  21473
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