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Theorem txcn 12481
 Description: A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
txcn.1 𝑋 = 𝑅
txcn.2 𝑌 = 𝑆
txcn.3 𝑍 = (𝑋 × 𝑌)
txcn.4 𝑊 = 𝑈
txcn.5 𝑃 = (1st𝑍)
txcn.6 𝑄 = (2nd𝑍)
Assertion
Ref Expression
txcn ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))

Proof of Theorem txcn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 txcn.1 . . . . 5 𝑋 = 𝑅
21toptopon 12222 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
3 txcn.2 . . . . 5 𝑌 = 𝑆
43toptopon 12222 . . . 4 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
5 txcn.5 . . . . . . 7 𝑃 = (1st𝑍)
6 txcn.3 . . . . . . . 8 𝑍 = (𝑋 × 𝑌)
76reseq2i 4823 . . . . . . 7 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
85, 7eqtri 2161 . . . . . 6 𝑃 = (1st ↾ (𝑋 × 𝑌))
9 tx1cn 12475 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
108, 9eqeltrid 2227 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
11 txcn.6 . . . . . . 7 𝑄 = (2nd𝑍)
126reseq2i 4823 . . . . . . 7 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
1311, 12eqtri 2161 . . . . . 6 𝑄 = (2nd ↾ (𝑋 × 𝑌))
14 tx2cn 12476 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2227 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
16 cnco 12427 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → (𝑃𝐹) ∈ (𝑈 Cn 𝑅))
17 cnco 12427 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝑄𝐹) ∈ (𝑈 Cn 𝑆))
1816, 17anim12dan 590 . . . . . 6 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ (𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)))
1918expcom 115 . . . . 5 ((𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
2010, 15, 19syl2anc 409 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
212, 4, 20syl2anb 289 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
22213adant3 1002 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
23 cntop1 12407 . . . . . . . 8 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
2423ad2antrl 482 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑈 ∈ Top)
25 txcn.4 . . . . . . . 8 𝑊 = 𝑈
2625topopn 12212 . . . . . . 7 (𝑈 ∈ Top → 𝑊𝑈)
2724, 26syl 14 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑊𝑈)
2825, 1cnf 12410 . . . . . . 7 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → (𝑃𝐹):𝑊𝑋)
2928ad2antrl 482 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑃𝐹):𝑊𝑋)
3025, 3cnf 12410 . . . . . . 7 ((𝑄𝐹) ∈ (𝑈 Cn 𝑆) → (𝑄𝐹):𝑊𝑌)
3130ad2antll 483 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑄𝐹):𝑊𝑌)
328, 13upxp 12478 . . . . . . 7 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
33 feq3 5264 . . . . . . . . . 10 (𝑍 = (𝑋 × 𝑌) → (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌)))
346, 33ax-mp 5 . . . . . . . . 9 (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌))
35343anbi1i 1173 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3635eubii 2009 . . . . . . 7 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3732, 36sylibr 133 . . . . . 6 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3827, 29, 31, 37syl3anc 1217 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
39 euex 2030 . . . . 5 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
4038, 39syl 14 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
41 simpll3 1023 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹:𝑊𝑍)
4227adantr 274 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑊𝑈)
431topopn 12212 . . . . . . . . . 10 (𝑅 ∈ Top → 𝑋𝑅)
443topopn 12212 . . . . . . . . . 10 (𝑆 ∈ Top → 𝑌𝑆)
45 xpexg 4660 . . . . . . . . . . 11 ((𝑋𝑅𝑌𝑆) → (𝑋 × 𝑌) ∈ V)
466, 45eqeltrid 2227 . . . . . . . . . 10 ((𝑋𝑅𝑌𝑆) → 𝑍 ∈ V)
4743, 44, 46syl2an 287 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 ∈ V)
48473adant3 1002 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 ∈ V)
4948ad2antrr 480 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑍 ∈ V)
50 fex2 5298 . . . . . . 7 ((𝐹:𝑊𝑍𝑊𝑈𝑍 ∈ V) → 𝐹 ∈ V)
5141, 42, 49, 50syl3anc 1217 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹 ∈ V)
52 eumo 2032 . . . . . . . 8 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
5338, 52syl 14 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
5453adantr 274 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
55 simpr 109 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
56 3anass 967 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
57 coeq2 4704 . . . . . . . . . . . 12 (𝐹 = → (𝑃𝐹) = (𝑃))
58 coeq2 4704 . . . . . . . . . . . 12 (𝐹 = → (𝑄𝐹) = (𝑄))
5957, 58jca 304 . . . . . . . . . . 11 (𝐹 = → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6059eqcoms 2143 . . . . . . . . . 10 ( = 𝐹 → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6160biantrud 302 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍 ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))))
62 feq1 5262 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍𝐹:𝑊𝑍))
6361, 62bitr3d 189 . . . . . . . 8 ( = 𝐹 → ((:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ↔ 𝐹:𝑊𝑍))
6456, 63syl5bb 191 . . . . . . 7 ( = 𝐹 → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ 𝐹:𝑊𝑍))
6564moi2 2868 . . . . . 6 (((𝐹 ∈ V ∧ ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ∧ ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ 𝐹:𝑊𝑍)) → = 𝐹)
6651, 54, 55, 41, 65syl22anc 1218 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → = 𝐹)
67 eqid 2140 . . . . . . . . . 10 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6867, 1, 3, 6, 5, 11uptx 12480 . . . . . . . . 9 (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6968adantl 275 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
70 df-reu 2424 . . . . . . . . . 10 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
71 euex 2030 . . . . . . . . . 10 (∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
7270, 71sylbi 120 . . . . . . . . 9 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
73 eqid 2140 . . . . . . . . . . . . . . 15 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7425, 73cnf 12410 . . . . . . . . . . . . . 14 ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊 (𝑅 ×t 𝑆))
751, 3txuni 12469 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
766, 75syl5eq 2185 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 = (𝑅 ×t 𝑆))
77763adant3 1002 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 = (𝑅 ×t 𝑆))
7877adantr 274 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑍 = (𝑅 ×t 𝑆))
7978feq3d 5268 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (:𝑊𝑍:𝑊 (𝑅 ×t 𝑆)))
8074, 79syl5ibr 155 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊𝑍))
8180anim1d 334 . . . . . . . . . . . 12 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))))
8281, 56syl6ibr 161 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
83 simpl 108 . . . . . . . . . . 11 (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
8482, 83jca2 306 . . . . . . . . . 10 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
8584eximdv 1853 . . . . . . . . 9 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∃((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
8672, 85syl5 32 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))))
8769, 86mpd 13 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
88 eupick 2079 . . . . . . 7 ((∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∃((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ ∈ (𝑈 Cn (𝑅 ×t 𝑆)))) → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
8938, 87, 88syl2anc 409 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
9089imp 123 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
9166, 90eqeltrrd 2218 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
9240, 91exlimddv 1871 . . 3 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
9392ex 114 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
9422, 93impbid 128 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃!weu 2000  ∃*wmo 2001  ∃!wreu 2419  Vcvv 2689  ∪ cuni 3743   × cxp 4544   ↾ cres 4548   ∘ ccom 4550  ⟶wf 5126  ‘cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  Topctop 12201  TopOnctopon 12214   Cn ccn 12391   ×t ctx 12458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-topgen 12178  df-top 12202  df-topon 12215  df-bases 12247  df-cn 12394  df-tx 12459 This theorem is referenced by: (None)
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