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Theorem txcnp 23568
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 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 txcnp.5 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝑌))
3 txcnp.8 . . . . . 6 (𝜑 → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
4 cnpf2 23198 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷)) → (𝑥𝑋𝐴):𝑋𝑌)
51, 2, 3, 4syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋𝑌)
65fvmptelcdm 7122 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
7 txcnp.6 . . . . . 6 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 txcnp.9 . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
9 cnpf2 23198 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑍) ∧ (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷)) → (𝑥𝑋𝐵):𝑋𝑍)
101, 7, 8, 9syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋𝑍)
1110fvmptelcdm 7122 . . . 4 ((𝜑𝑥𝑋) → 𝐵𝑍)
126, 11opelxpd 5717 . . 3 ((𝜑𝑥𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑌 × 𝑍))
1312fmpttd 7124 . 2 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩):𝑋⟶(𝑌 × 𝑍))
14 txcnp.7 . . . . . . . . 9 (𝜑𝐷𝑋)
15 simpr 483 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → 𝑥𝑋)
16 opex 5466 . . . . . . . . . . . 12 𝐴, 𝐵⟩ ∈ V
17 eqid 2725 . . . . . . . . . . . . 13 (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)
1817fvmpt2 7015 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
1915, 16, 18sylancl 584 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨𝐴, 𝐵⟩)
20 eqid 2725 . . . . . . . . . . . . . 14 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
2120fvmpt2 7015 . . . . . . . . . . . . 13 ((𝑥𝑋𝐴𝑌) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
2215, 6, 21syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐴)‘𝑥) = 𝐴)
23 eqid 2725 . . . . . . . . . . . . . 14 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
2423fvmpt2 7015 . . . . . . . . . . . . 13 ((𝑥𝑋𝐵𝑍) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
2515, 11, 24syl2anc 582 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ((𝑥𝑋𝐵)‘𝑥) = 𝐵)
2622, 25opeq12d 4883 . . . . . . . . . . 11 ((𝜑𝑥𝑋) → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
2719, 26eqtr4d 2768 . . . . . . . . . 10 ((𝜑𝑥𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
2827ralrimiva 3135 . . . . . . . . 9 (𝜑 → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
29 nffvmpt1 6907 . . . . . . . . . . 11 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷)
30 nffvmpt1 6907 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐴)‘𝐷)
31 nffvmpt1 6907 . . . . . . . . . . . 12 𝑥((𝑥𝑋𝐵)‘𝐷)
3230, 31nfop 4891 . . . . . . . . . . 11 𝑥⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
3329, 32nfeq 2905 . . . . . . . . . 10 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩
34 fveq2 6896 . . . . . . . . . . 11 (𝑥 = 𝐷 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷))
35 fveq2 6896 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝐷))
36 fveq2 6896 . . . . . . . . . . . 12 (𝑥 = 𝐷 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝐷))
3735, 36opeq12d 4883 . . . . . . . . . . 11 (𝑥 = 𝐷 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
3834, 37eqeq12d 2741 . . . . . . . . . 10 (𝑥 = 𝐷 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
3933, 38rspc 3594 . . . . . . . . 9 (𝐷𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩))
4014, 28, 39sylc 65 . . . . . . . 8 (𝜑 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) = ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩)
4140eleq1d 2810 . . . . . . 7 (𝜑 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
4241adantr 479 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) ↔ ⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤)))
433ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))
44 simplrl 775 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑣𝐾)
45 simprl 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣)
46 cnpimaex 23204 . . . . . . . . . 10 (((𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷) ∧ 𝑣𝐾 ∧ ((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
4743, 44, 45, 46syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣))
488ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))
49 simplrr 776 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → 𝑤𝐿)
50 simprr 771 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)
51 cnpimaex 23204 . . . . . . . . . 10 (((𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷) ∧ 𝑤𝐿 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5248, 49, 50, 51syl3anc 1368 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))
5347, 52jca 510 . . . . . . . 8 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤)) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
5453ex 411 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤) → (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
55 opelxp 5714 . . . . . . 7 (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) ↔ (((𝑥𝑋𝐴)‘𝐷) ∈ 𝑣 ∧ ((𝑥𝑋𝐵)‘𝐷) ∈ 𝑤))
56 reeanv 3216 . . . . . . 7 (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ (∃𝑟𝐽 (𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ ∃𝑠𝐽 (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
5754, 55, 563imtr4g 295 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (⟨((𝑥𝑋𝐴)‘𝐷), ((𝑥𝑋𝐵)‘𝐷)⟩ ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
5842, 57sylbid 239 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))))
59 an4 654 . . . . . . . . . . 11 (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) ↔ ((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)))
60 elin 3960 . . . . . . . . . . . . . 14 (𝐷 ∈ (𝑟𝑠) ↔ (𝐷𝑟𝐷𝑠))
6160biimpri 227 . . . . . . . . . . . . 13 ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠))
6261a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝐷𝑟𝐷𝑠) → 𝐷 ∈ (𝑟𝑠)))
63 simpl 481 . . . . . . . . . . . . . . . 16 ((𝑟𝐽𝑠𝐽) → 𝑟𝐽)
64 toponss 22873 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑟𝐽) → 𝑟𝑋)
651, 63, 64syl2an 594 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → 𝑟𝑋)
66 ssinss1 4236 . . . . . . . . . . . . . . . . . . . . 21 (𝑟𝑋 → (𝑟𝑠) ⊆ 𝑋)
6766adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ 𝑋)
6867sselda 3976 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑋)
6928ad2antrr 724 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩)
70 nffvmpt1 6907 . . . . . . . . . . . . . . . . . . . . 21 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡)
71 nffvmpt1 6907 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐴)‘𝑡)
72 nffvmpt1 6907 . . . . . . . . . . . . . . . . . . . . . 22 𝑥((𝑥𝑋𝐵)‘𝑡)
7371, 72nfop 4891 . . . . . . . . . . . . . . . . . . . . 21 𝑥⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
7470, 73nfeq 2905 . . . . . . . . . . . . . . . . . . . 20 𝑥((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩
75 fveq2 6896 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡))
76 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐴)‘𝑥) = ((𝑥𝑋𝐴)‘𝑡))
77 fveq2 6896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑡 → ((𝑥𝑋𝐵)‘𝑥) = ((𝑥𝑋𝐵)‘𝑡))
7876, 77opeq12d 4883 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑡 → ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
7975, 78eqeq12d 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑡 → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8074, 79rspc 3594 . . . . . . . . . . . . . . . . . . 19 (𝑡𝑋 → (∀𝑥𝑋 ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑥) = ⟨((𝑥𝑋𝐴)‘𝑥), ((𝑥𝑋𝐵)‘𝑥)⟩ → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩))
8168, 69, 80sylc 65 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) = ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩)
82 simpr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ (𝑟𝑠))
8382elin1d 4196 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑟)
845ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐴):𝑋𝑌)
8584ffund 6727 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐴))
8667adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ 𝑋)
8784fdmd 6733 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐴) = 𝑋)
8886, 87sseqtrrd 4018 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐴))
8988, 82sseldd 3977 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐴))
90 funfvima 7242 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐴) ∧ 𝑡 ∈ dom (𝑥𝑋𝐴)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
9185, 89, 90syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑟 → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟)))
9283, 91mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐴)‘𝑡) ∈ ((𝑥𝑋𝐴) “ 𝑟))
9382elin2d 4197 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡𝑠)
9410ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑥𝑋𝐵):𝑋𝑍)
9594ffund 6727 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → Fun (𝑥𝑋𝐵))
9694fdmd 6733 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → dom (𝑥𝑋𝐵) = 𝑋)
9786, 96sseqtrrd 4018 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑟𝑠) ⊆ dom (𝑥𝑋𝐵))
9897, 82sseldd 3977 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → 𝑡 ∈ dom (𝑥𝑋𝐵))
99 funfvima 7242 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (𝑥𝑋𝐵) ∧ 𝑡 ∈ dom (𝑥𝑋𝐵)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
10095, 98, 99syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → (𝑡𝑠 → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠)))
10193, 100mpd 15 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋𝐵)‘𝑡) ∈ ((𝑥𝑋𝐵) “ 𝑠))
10292, 101opelxpd 5717 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ⟨((𝑥𝑋𝐴)‘𝑡), ((𝑥𝑋𝐵)‘𝑡)⟩ ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
10381, 102eqeltrd 2825 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟𝑋) ∧ 𝑡 ∈ (𝑟𝑠)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
104103ralrimiva 3135 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
10513ffund 6727 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
106105adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → Fun (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
10713fdmd 6733 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
108107adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟𝑋) → dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) = 𝑋)
10967, 108sseqtrrd 4018 . . . . . . . . . . . . . . . . 17 ((𝜑𝑟𝑋) → (𝑟𝑠) ⊆ dom (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩))
110 funimass4 6962 . . . . . . . . . . . . . . . . 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))))
111106, 109, 110syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝜑𝑟𝑋) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ↔ ∀𝑡 ∈ (𝑟𝑠)((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝑡) ∈ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠))))
112104, 111mpbird 256 . . . . . . . . . . . . . . 15 ((𝜑𝑟𝑋) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
11365, 112syldan 589 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
114113adantlr 713 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)))
115 xpss12 5693 . . . . . . . . . . . . 13 ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤))
116 sstr2 3983 . . . . . . . . . . . . 13 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) → ((((𝑥𝑋𝐴) “ 𝑟) × ((𝑥𝑋𝐵) “ 𝑠)) ⊆ (𝑣 × 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
117114, 115, 116syl2im 40 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → ((((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
11862, 117anim12d 607 . . . . . . . . . . 11 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟𝐷𝑠) ∧ (((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
11959, 118biimtrid 241 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
120 topontop 22859 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1211, 120syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
122 inopn 22845 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑟𝐽𝑠𝐽) → (𝑟𝑠) ∈ 𝐽)
1231223expb 1117 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
124121, 123sylan 578 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
125124adantlr 713 . . . . . . . . . 10 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (𝑟𝑠) ∈ 𝐽)
126119, 125jctild 524 . . . . . . . . 9 (((𝜑 ∧ (𝑣𝐾𝑤𝐿)) ∧ (𝑟𝐽𝑠𝐽)) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
127126expimpd 452 . . . . . . . 8 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))))
128 eleq2 2814 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (𝐷𝑧𝐷 ∈ (𝑟𝑠)))
129 imaeq2 6060 . . . . . . . . . . 11 (𝑧 = (𝑟𝑠) → ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) = ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)))
130129sseq1d 4008 . . . . . . . . . 10 (𝑧 = (𝑟𝑠) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤) ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤)))
131128, 130anbi12d 630 . . . . . . . . 9 (𝑧 = (𝑟𝑠) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)) ↔ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))))
132131rspcev 3606 . . . . . . . 8 (((𝑟𝑠) ∈ 𝐽 ∧ (𝐷 ∈ (𝑟𝑠) ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ (𝑟𝑠)) ⊆ (𝑣 × 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
133127, 132syl6 35 . . . . . . 7 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑟𝐽𝑠𝐽) ∧ ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤))) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
134133expd 414 . . . . . 6 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → ((𝑟𝐽𝑠𝐽) → (((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))))
135134rexlimdvv 3200 . . . . 5 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (∃𝑟𝐽𝑠𝐽 ((𝐷𝑟 ∧ ((𝑥𝑋𝐴) “ 𝑟) ⊆ 𝑣) ∧ (𝐷𝑠 ∧ ((𝑥𝑋𝐵) “ 𝑠) ⊆ 𝑤)) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
13658, 135syld 47 . . . 4 ((𝜑 ∧ (𝑣𝐾𝑤𝐿)) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
137136ralrimivva 3190 . . 3 (𝜑 → ∀𝑣𝐾𝑤𝐿 (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
138 vex 3465 . . . . . 6 𝑣 ∈ V
139 vex 3465 . . . . . 6 𝑤 ∈ V
140138, 139xpex 7756 . . . . 5 (𝑣 × 𝑤) ∈ V
141140rgen2w 3055 . . . 4 𝑣𝐾𝑤𝐿 (𝑣 × 𝑤) ∈ V
142 eqid 2725 . . . . 5 (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
143 eleq2 2814 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ (𝑣 × 𝑤)))
144 sseq2 4003 . . . . . . . 8 (𝑦 = (𝑣 × 𝑤) → (((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦 ↔ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤)))
145144anbi2d 628 . . . . . . 7 (𝑦 = (𝑣 × 𝑤) → ((𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
146145rexbidv 3168 . . . . . 6 (𝑦 = (𝑣 × 𝑤) → (∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ (𝑣 × 𝑤))))
147143, 146imbi12d 343 . . . . 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)))))
148142, 147ralrnmpo 7560 . . . 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)))))
149141, 148ax-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))))
150137, 149sylibr 233 . 2 (𝜑 → ∀𝑦 ∈ ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))(((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)‘𝐷) ∈ 𝑦 → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) “ 𝑧) ⊆ 𝑦)))
151 topontop 22859 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
1522, 151syl 17 . . . 4 (𝜑𝐾 ∈ Top)
153 topontop 22859 . . . . 5 (𝐿 ∈ (TopOn‘𝑍) → 𝐿 ∈ Top)
1547, 153syl 17 . . . 4 (𝜑𝐿 ∈ Top)
155 eqid 2725 . . . . 5 ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤)) = ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))
156155txval 23512 . . . 4 ((𝐾 ∈ Top ∧ 𝐿 ∈ Top) → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
157152, 154, 156syl2anc 582 . . 3 (𝜑 → (𝐾 ×t 𝐿) = (topGen‘ran (𝑣𝐾, 𝑤𝐿 ↦ (𝑣 × 𝑤))))
158 txtopon 23539 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (TopOn‘𝑍)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1592, 7, 158syl2anc 582 . . 3 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑌 × 𝑍)))
1601, 157, 159, 14tgcnp 23201 . 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)))))
16113, 150, 160mpbir2and 711 1 (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944  cop 4636  cmpt 5232   × cxp 5676  dom cdm 5678  ran crn 5679  cima 5681  Fun wfun 6543  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  topGenctg 17422  Topctop 22839  TopOnctopon 22856   CnP ccnp 23173   ×t ctx 23508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-topgen 17428  df-top 22840  df-topon 22857  df-bases 22893  df-cnp 23176  df-tx 23510
This theorem is referenced by:  limccnp2  25865
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