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Theorem txcnp 21333
Description: If two functions are continuous at 𝐷, then the ordered pair of them is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
txcnp.4 (𝜑𝐽 ∈ (TopOn‘𝑋))
txcnp.5 (𝜑𝐾 ∈ (TopOn‘𝑌))
txcnp.6 (𝜑𝐿 ∈ (TopOn‘𝑍))
txcnp.7 (𝜑𝐷𝑋)
txcnp.8 (𝜑 → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
txcnp.9 (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
Assertion
Ref Expression
txcnp (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
Distinct variable groups:   𝜑,𝑥   𝑥,𝑌   𝑥,𝑍   𝑥,𝐷   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐽(𝑥)   𝐾(𝑥)   𝐿(𝑥)

Proof of Theorem txcnp
Dummy variables 𝑠 𝑟 𝑡 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txcnp.4 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 txcnp.5 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 txcnp.8 . . . . . . 7 (𝜑 → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
4 cnpf2 20964 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1323 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
6 eqid 2621 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6337 . . . . . 6 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
85, 7sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
98r19.21bi 2927 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
10 txcnp.6 . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
11 txcnp.9 . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
12 cnpf2 20964 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥𝑋𝐵):𝑋𝑍)
131, 10, 11, 12syl3anc 1323 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
14 eqid 2621 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1514fmpt 6337 . . . . . 6 (∀𝑥𝑋 𝐵𝑍 ↔ (𝑥𝑋𝐵):𝑋𝑍)
1613, 15sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵𝑍)
1716r19.21bi 2927 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
18 opelxpi 5108 . . . 4 ((𝐴𝑌𝐵𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
199, 17, 18syl2anc 692 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
20 eqid 2621 . . 3 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
2119, 20fmptd 6340 . 2 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍))
22 txcnp.7 . . . . . . . . 9 (𝜑𝐷𝑋)
23 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝑥𝑋)
24 opex 4893 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
2520fvmpt2 6248 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
2623, 24, 25sylancl 693 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
276fvmpt2 6248 . . . . . . . . . . . . 13 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2823, 9, 27syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2914fvmpt2 6248 . . . . . . . . . . . . 13 ((𝑥𝑋𝐵𝑍) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3023, 17, 29syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3128, 30opeq12d 4378 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3226, 31eqtr4d 2658 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
3332ralrimiva 2960 . . . . . . . . 9 (𝜑 → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
34 nffvmpt1 6156 . . . . . . . . . . 11 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷)
35 nffvmpt1 6156 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐴)‘𝐷)
36 nffvmpt1 6156 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐵)‘𝐷)
3735, 36nfop 4386 . . . . . . . . . . 11 𝑥⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
3834, 37nfeq 2772 . . . . . . . . . 10 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
39 fveq2 6148 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷))
40 fveq2 6148 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
41 fveq2 6148 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝐷))
4240, 41opeq12d 4378 . . . . . . . . . . 11 (𝑥 = 𝐷 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
4339, 42eqeq12d 2636 . . . . . . . . . 10 (𝑥 = 𝐷 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
4438, 43rspc 3289 . . . . . . . . 9 (𝐷𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
4522, 33, 44sylc 65 . . . . . . . 8 (𝜑 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
4645eleq1d 2683 . . . . . . 7 (𝜑 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
4746adantr 481 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
483ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
49 simplrl 799 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑣𝐾)
50 simprl 793 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣)
51 cnpimaex 20970 . . . . . . . . . 10 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣𝐾 ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
5248, 49, 50, 51syl3anc 1323 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
5311ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
54 simplrr 800 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑤𝐿)
55 simprr 795 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)
56 cnpimaex 20970 . . . . . . . . . 10 (((𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤𝐿 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5753, 54, 55, 56syl3anc 1323 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5852, 57jca 554 . . . . . . . 8 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
5958ex 450 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
60 opelxp 5106 . . . . . . 7 (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) ↔ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤))
61 reeanv 3097 . . . . . . 7 (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
6259, 60, 613imtr4g 285 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
6347, 62sylbid 230 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
64 an4 864 . . . . . . . . . . 11 (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ ((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
65 elin 3774 . . . . . . . . . . . . . 14 (𝐷 ∈ (𝑟𝑠) ↔ (𝐷𝑟𝐷𝑠))
6665biimpri 218 . . . . . . . . . . . . 13 ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠))
6766a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠)))
68 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑟𝐽𝑠𝐽) → 𝑟𝐽)
69 toponss 20644 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟𝐽) → 𝑟𝑋)
701, 68, 69syl2an 494 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → 𝑟𝑋)
71 ssinss1 3819 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑋 → (𝑟𝑠) ⊆ 𝑋)
7271adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ 𝑋)
7372sselda 3583 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑋)
7433ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
75 nffvmpt1 6156 . . . . . . . . . . . . . . . . . . . . 21 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡)
76 nffvmpt1 6156 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐴)‘𝑡)
77 nffvmpt1 6156 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐵)‘𝑡)
7876, 77nfop 4386 . . . . . . . . . . . . . . . . . . . . 21 𝑥⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
7975, 78nfeq 2772 . . . . . . . . . . . . . . . . . . . 20 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
80 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡))
81 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑡))
82 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑡))
8381, 82opeq12d 4378 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
8480, 83eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8579, 84rspc 3289 . . . . . . . . . . . . . . . . . . 19 (𝑡𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8673, 74, 85sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
87 inss1 3811 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑠) ⊆ 𝑟
88 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ (𝑟𝑠))
8987, 88sseldi 3581 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑟)
905ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐴):𝑋𝑌)
91 ffun 6005 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝐴):𝑋𝑌 → Fun (𝑥𝑋𝐴))
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐴))
9372adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ 𝑋)
94 fdm 6008 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝑋𝐴):𝑋𝑌 → dom (𝑥𝑋𝐴) = 𝑋)
9590, 94syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐴) = 𝑋)
9693, 95sseqtr4d 3621 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐴))
9796, 88sseldd 3584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐴))
98 funfvima 6446 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐴) ∧ 𝑡 ∈ dom (𝑥𝑋𝐴)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
9992, 97, 98syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
10089, 99mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟))
101 inss2 3812 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑠) ⊆ 𝑠
102101, 88sseldi 3581 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑠)
10313ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐵):𝑋𝑍)
104 ffun 6005 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝐵):𝑋𝑍 → Fun (𝑥𝑋𝐵))
105103, 104syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐵))
106 fdm 6008 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝑋𝐵):𝑋𝑍 → dom (𝑥𝑋𝐵) = 𝑋)
107103, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐵) = 𝑋)
10893, 107sseqtr4d 3621 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐵))
109108, 88sseldd 3584 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐵))
110 funfvima 6446 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐵) ∧ 𝑡 ∈ dom (𝑥𝑋𝐵)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
111105, 109, 110syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
112102, 111mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠))
113 opelxpi 5108 . . . . . . . . . . . . . . . . . . 19 ((((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟) ∧ ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)) → ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩ ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
114100, 112, 113syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩ ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
11586, 114eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
116115ralrimiva 2960 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
117 ffun 6005 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
11821, 117syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
119118adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
120 fdm 6008 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
12121, 120syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
122121adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟𝑋) → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
12372, 122sseqtr4d 3621 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
124 funimass4 6204 . . . . . . . . . . . . . . . . 17 ((Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∧ (𝑟𝑠) ⊆ dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠))))
125119, 123, 124syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠))))
126116, 125mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑𝑟𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
12770, 126syldan 487 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
128127adantlr 750 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
129 xpss12 5186 . . . . . . . . . . . . 13 ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤))
130 sstr2 3590 . . . . . . . . . . . . 13 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) → ((((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
131128, 129, 130syl2im 40 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
13267, 131anim12d 585 . . . . . . . . . . 11 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
13364, 132syl5bi 232 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
134 topontop 20641 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1351, 134syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
136 inopn 20629 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑟𝐽𝑠𝐽) → (𝑟𝑠) ∈ 𝐽)
1371363expb 1263 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
138135, 137sylan 488 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
139138adantlr 750 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
140133, 139jctild 565 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
141140expimpd 628 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
142 eleq2 2687 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (𝐷𝑧𝐷 ∈ (𝑟𝑠)))
143 imaeq2 5421 . . . . . . . . . . 11 (𝑧 = (𝑟𝑠) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)))
144143sseq1d 3611 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
145142, 144anbi12d 746 . . . . . . . . 9 (𝑧 = (𝑟𝑠) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
146145rspcev 3295 . . . . . . . 8 (((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
147141, 146syl6 35 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
148147expd 452 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((𝑟𝐽𝑠𝐽) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
149148rexlimdvv 3030 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
15063, 149syld 47 . . . 4 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
151150ralrimivva 2965 . . 3 (𝜑 → ∀𝑣𝐾𝑤𝐿 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
152 vex 3189 . . . . . 6 𝑣 ∈ V
153 vex 3189 . . . . . 6 𝑤 ∈ V
154152, 153xpex 6915 . . . . 5 (𝑣 × 𝑤) ∈ V
155154rgen2w 2920 . . . 4 𝑣𝐾𝑤𝐿 (𝑣 × 𝑤) ∈ V
156 eqid 2621 . . . . 5 (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
157 eleq2 2687 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤)))
158 sseq2 3606 . . . . . . . 8 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
159158anbi2d 739 . . . . . . 7 (𝑦 = (𝑣 × 𝑤) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
160159rexbidv 3045 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
161157, 160imbi12d 334 . . . . 5 (𝑦 = (𝑣 × 𝑤) → ((((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
162156, 161ralrnmpt2 6728 . . . 4 (∀𝑣𝐾𝑤𝐿 (𝑣 × 𝑤) ∈ V → (∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣𝐾𝑤𝐿 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
163155, 162ax-mp 5 . . 3 (∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)) ↔ ∀𝑣𝐾𝑤𝐿 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
164151, 163sylibr 224 . 2 (𝜑 → ∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))
165 topontop 20641 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1662, 165syl 17 . . . 4 (𝜑𝐾 ∈ Top)
167 topontop 20641 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
16810, 167syl 17 . . . 4 (𝜑𝐿 ∈ Top)
169 eqid 2621 . . . . 5 ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
170169txval 21277 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
171166, 168, 170syl2anc 692 . . 3 (𝜑 → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
172 txtopon 21304 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1732, 10, 172syl2anc 692 . . 3 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1741, 171, 173, 22tgcnp 20967 . 2 (𝜑 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷) ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) ∧ ∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))))
17521, 164, 174mpbir2and 956 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  cop 4154  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  topGenctg 16019  Topctop 20617  TopOnctopon 20618   CnP ccnp 20939   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cnp 20942  df-tx 21275
This theorem is referenced by:  limccnp2  23562
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