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Theorem txcnp 21233
 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 20864 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
6 eqid 2610 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
76fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐴𝑌 ↔ (𝑥𝑋𝐴):𝑋𝑌)
85, 7sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
98r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
10 txcnp.6 . . . . . . 7 (𝜑𝐿 ∈ (TopOn‘𝑍))
11 txcnp.9 . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
12 cnpf2 20864 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥𝑋𝐵):𝑋𝑍)
131, 10, 11, 12syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
14 eqid 2610 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1514fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐵𝑍 ↔ (𝑥𝑋𝐵):𝑋𝑍)
1613, 15sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵𝑍)
1716r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
18 opelxpi 5072 . . . 4 ((𝐴𝑌𝐵𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
199, 17, 18syl2anc 691 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
20 eqid 2610 . . 3 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
2119, 20fmptd 6292 . 2 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍))
22 txcnp.7 . . . . . . . . 9 (𝜑𝐷𝑋)
23 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝑥𝑋)
24 opex 4859 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
2520fvmpt2 6200 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
2623, 24, 25sylancl 693 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
276fvmpt2 6200 . . . . . . . . . . . . 13 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2823, 9, 27syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2914fvmpt2 6200 . . . . . . . . . . . . 13 ((𝑥𝑋𝐵𝑍) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3023, 17, 29syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
3128, 30opeq12d 4348 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
3226, 31eqtr4d 2647 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
3332ralrimiva 2949 . . . . . . . . 9 (𝜑 → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
34 nffvmpt1 6111 . . . . . . . . . . 11 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷)
35 nffvmpt1 6111 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐴)‘𝐷)
36 nffvmpt1 6111 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐵)‘𝐷)
3735, 36nfop 4356 . . . . . . . . . . 11 𝑥⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
3834, 37nfeq 2762 . . . . . . . . . 10 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
39 fveq2 6103 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷))
40 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
41 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝐷))
4240, 41opeq12d 4348 . . . . . . . . . . 11 (𝑥 = 𝐷 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
4339, 42eqeq12d 2625 . . . . . . . . . 10 (𝑥 = 𝐷 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
4438, 43rspc 3276 . . . . . . . . 9 (𝐷𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
4522, 33, 44sylc 63 . . . . . . . 8 (𝜑 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
4645eleq1d 2672 . . . . . . 7 (𝜑 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
4746adantr 480 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
483ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
49 simplrl 796 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑣𝐾)
50 simprl 790 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣)
51 cnpimaex 20870 . . . . . . . . . 10 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣𝐾 ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
5248, 49, 50, 51syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
5311ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
54 simplrr 797 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑤𝐿)
55 simprr 792 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)
56 cnpimaex 20870 . . . . . . . . . 10 (((𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤𝐿 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5753, 54, 55, 56syl3anc 1318 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5852, 57jca 553 . . . . . . . 8 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
5958ex 449 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
60 opelxp 5070 . . . . . . 7 (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) ↔ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤))
61 reeanv 3086 . . . . . . 7 (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
6259, 60, 613imtr4g 284 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
6347, 62sylbid 229 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
64 an4 861 . . . . . . . . . . 11 (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ ((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
65 elin 3758 . . . . . . . . . . . . . 14 (𝐷 ∈ (𝑟𝑠) ↔ (𝐷𝑟𝐷𝑠))
6665biimpri 217 . . . . . . . . . . . . 13 ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠))
6766a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠)))
68 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑟𝐽𝑠𝐽) → 𝑟𝐽)
69 toponss 20544 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟𝐽) → 𝑟𝑋)
701, 68, 69syl2an 493 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → 𝑟𝑋)
71 ssinss1 3803 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑋 → (𝑟𝑠) ⊆ 𝑋)
7271adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ 𝑋)
7372sselda 3568 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑋)
7433ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
75 nffvmpt1 6111 . . . . . . . . . . . . . . . . . . . . 21 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡)
76 nffvmpt1 6111 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐴)‘𝑡)
77 nffvmpt1 6111 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐵)‘𝑡)
7876, 77nfop 4356 . . . . . . . . . . . . . . . . . . . . 21 𝑥⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
7975, 78nfeq 2762 . . . . . . . . . . . . . . . . . . . 20 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
80 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡))
81 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑡))
82 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑡))
8381, 82opeq12d 4348 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
8480, 83eqeq12d 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8579, 84rspc 3276 . . . . . . . . . . . . . . . . . . 19 (𝑡𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8673, 74, 85sylc 63 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
87 inss1 3795 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑠) ⊆ 𝑟
88 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ (𝑟𝑠))
8987, 88sseldi 3566 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑟)
905ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐴):𝑋𝑌)
91 ffun 5961 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝐴):𝑋𝑌 → Fun (𝑥𝑋𝐴))
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐴))
9372adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ 𝑋)
94 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝑋𝐴):𝑋𝑌 → dom (𝑥𝑋𝐴) = 𝑋)
9590, 94syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐴) = 𝑋)
9693, 95sseqtr4d 3605 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐴))
9796, 88sseldd 3569 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐴))
98 funfvima 6396 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐴) ∧ 𝑡 ∈ dom (𝑥𝑋𝐴)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
9992, 97, 98syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
10089, 99mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟))
101 inss2 3796 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑠) ⊆ 𝑠
102101, 88sseldi 3566 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑠)
10313ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐵):𝑋𝑍)
104 ffun 5961 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝐵):𝑋𝑍 → Fun (𝑥𝑋𝐵))
105103, 104syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐵))
106 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥𝑋𝐵):𝑋𝑍 → dom (𝑥𝑋𝐵) = 𝑋)
107103, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐵) = 𝑋)
10893, 107sseqtr4d 3605 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐵))
109108, 88sseldd 3569 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐵))
110 funfvima 6396 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐵) ∧ 𝑡 ∈ dom (𝑥𝑋𝐵)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
111105, 109, 110syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
112102, 111mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠))
113 opelxpi 5072 . . . . . . . . . . . . . . . . . . 19 ((((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟) ∧ ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)) → ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩ ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
114100, 112, 113syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩ ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
11586, 114eqeltrd 2688 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
116115ralrimiva 2949 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
117 ffun 5961 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
11821, 117syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
119118adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
120 fdm 5964 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍) → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
12121, 120syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
122121adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟𝑋) → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
12372, 122sseqtr4d 3605 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
124 funimass4 6157 . . . . . . . . . . . . . . . . 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 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠))))
126116, 125mpbird 246 . . . . . . . . . . . . . . 15 ((𝜑𝑟𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
12770, 126syldan 486 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
128127adantlr 747 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
129 xpss12 5148 . . . . . . . . . . . . 13 ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤))
130 sstr2 3575 . . . . . . . . . . . . 13 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) → ((((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
131128, 129, 130syl2im 39 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
13267, 131anim12d 584 . . . . . . . . . . 11 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
13364, 132syl5bi 231 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
134 topontop 20541 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1351, 134syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
136 inopn 20529 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑟𝐽𝑠𝐽) → (𝑟𝑠) ∈ 𝐽)
1371363expb 1258 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
138135, 137sylan 487 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
139138adantlr 747 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
140133, 139jctild 564 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
141140expimpd 627 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
142 eleq2 2677 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (𝐷𝑧𝐷 ∈ (𝑟𝑠)))
143 imaeq2 5381 . . . . . . . . . . 11 (𝑧 = (𝑟𝑠) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)))
144143sseq1d 3595 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
145142, 144anbi12d 743 . . . . . . . . 9 (𝑧 = (𝑟𝑠) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
146145rspcev 3282 . . . . . . . 8 (((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
147141, 146syl6 34 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
148147expd 451 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((𝑟𝐽𝑠𝐽) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
149148rexlimdvv 3019 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
15063, 149syld 46 . . . 4 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
151150ralrimivva 2954 . . 3 (𝜑 → ∀𝑣𝐾𝑤𝐿 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
152 vex 3176 . . . . . 6 𝑣 ∈ V
153 vex 3176 . . . . . 6 𝑤 ∈ V
154152, 153xpex 6860 . . . . 5 (𝑣 × 𝑤) ∈ V
155154rgen2w 2909 . . . 4 𝑣𝐾𝑤𝐿 (𝑣 × 𝑤) ∈ V
156 eqid 2610 . . . . 5 (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
157 eleq2 2677 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤)))
158 sseq2 3590 . . . . . . . 8 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
159158anbi2d 736 . . . . . . 7 (𝑦 = (𝑣 × 𝑤) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
160159rexbidv 3034 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
161157, 160imbi12d 333 . . . . 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 6673 . . . 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 223 . 2 (𝜑 → ∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))
165 topontop 20541 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1662, 165syl 17 . . . 4 (𝜑𝐾 ∈ Top)
167 topontop 20541 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
16810, 167syl 17 . . . 4 (𝜑𝐿 ∈ Top)
169 eqid 2610 . . . . 5 ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
170169txval 21177 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
171166, 168, 170syl2anc 691 . . 3 (𝜑 → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
172 txtopon 21204 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1732, 10, 172syl2anc 691 . . 3 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1741, 171, 173, 22tgcnp 20867 . 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 959 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  topGenctg 15921  Topctop 20517  TopOnctopon 20518   CnP ccnp 20839   ×t ctx 21173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cnp 20842  df-tx 21175 This theorem is referenced by:  limccnp2  23462
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