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Theorem txcn 13442
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 13183 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
3 txcn.2 . . . . 5 𝑌 = 𝑆
43toptopon 13183 . . . 4 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
5 txcn.5 . . . . . . 7 𝑃 = (1st𝑍)
6 txcn.3 . . . . . . . 8 𝑍 = (𝑋 × 𝑌)
76reseq2i 4900 . . . . . . 7 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
85, 7eqtri 2198 . . . . . 6 𝑃 = (1st ↾ (𝑋 × 𝑌))
9 tx1cn 13436 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
108, 9eqeltrid 2264 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
11 txcn.6 . . . . . . 7 𝑄 = (2nd𝑍)
126reseq2i 4900 . . . . . . 7 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
1311, 12eqtri 2198 . . . . . 6 𝑄 = (2nd ↾ (𝑋 × 𝑌))
14 tx2cn 13437 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2264 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
16 cnco 13388 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → (𝑃𝐹) ∈ (𝑈 Cn 𝑅))
17 cnco 13388 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝑄𝐹) ∈ (𝑈 Cn 𝑆))
1816, 17anim12dan 600 . . . . . 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 116 . . . . 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 411 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
212, 4, 20syl2anb 291 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
22213adant3 1017 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
23 cntop1 13368 . . . . . . . 8 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
2423ad2antrl 490 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑈 ∈ Top)
25 txcn.4 . . . . . . . 8 𝑊 = 𝑈
2625topopn 13173 . . . . . . 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 13371 . . . . . . 7 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → (𝑃𝐹):𝑊𝑋)
2928ad2antrl 490 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑃𝐹):𝑊𝑋)
3025, 3cnf 13371 . . . . . . 7 ((𝑄𝐹) ∈ (𝑈 Cn 𝑆) → (𝑄𝐹):𝑊𝑌)
3130ad2antll 491 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑄𝐹):𝑊𝑌)
328, 13upxp 13439 . . . . . . 7 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
33 feq3 5346 . . . . . . . . . 10 (𝑍 = (𝑋 × 𝑌) → (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌)))
346, 33ax-mp 5 . . . . . . . . 9 (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌))
35343anbi1i 1190 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3635eubii 2035 . . . . . . 7 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3732, 36sylibr 134 . . . . . 6 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3827, 29, 31, 37syl3anc 1238 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
39 euex 2056 . . . . 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 1038 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹:𝑊𝑍)
4227adantr 276 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑊𝑈)
431topopn 13173 . . . . . . . . . 10 (𝑅 ∈ Top → 𝑋𝑅)
443topopn 13173 . . . . . . . . . 10 (𝑆 ∈ Top → 𝑌𝑆)
45 xpexg 4737 . . . . . . . . . . 11 ((𝑋𝑅𝑌𝑆) → (𝑋 × 𝑌) ∈ V)
466, 45eqeltrid 2264 . . . . . . . . . 10 ((𝑋𝑅𝑌𝑆) → 𝑍 ∈ V)
4743, 44, 46syl2an 289 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 ∈ V)
48473adant3 1017 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 ∈ V)
4948ad2antrr 488 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑍 ∈ V)
50 fex2 5380 . . . . . . 7 ((𝐹:𝑊𝑍𝑊𝑈𝑍 ∈ V) → 𝐹 ∈ V)
5141, 42, 49, 50syl3anc 1238 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹 ∈ V)
52 eumo 2058 . . . . . . . 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 276 . . . . . 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 110 . . . . . 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 982 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
57 coeq2 4781 . . . . . . . . . . . 12 (𝐹 = → (𝑃𝐹) = (𝑃))
58 coeq2 4781 . . . . . . . . . . . 12 (𝐹 = → (𝑄𝐹) = (𝑄))
5957, 58jca 306 . . . . . . . . . . 11 (𝐹 = → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6059eqcoms 2180 . . . . . . . . . 10 ( = 𝐹 → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6160biantrud 304 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍 ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))))
62 feq1 5344 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍𝐹:𝑊𝑍))
6361, 62bitr3d 190 . . . . . . . 8 ( = 𝐹 → ((:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ↔ 𝐹:𝑊𝑍))
6456, 63bitrid 192 . . . . . . 7 ( = 𝐹 → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ 𝐹:𝑊𝑍))
6564moi2 2918 . . . . . 6 (((𝐹 ∈ V ∧ ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ∧ ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ 𝐹:𝑊𝑍)) → = 𝐹)
6651, 54, 55, 41, 65syl22anc 1239 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → = 𝐹)
67 eqid 2177 . . . . . . . . . 10 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6867, 1, 3, 6, 5, 11uptx 13441 . . . . . . . . 9 (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6968adantl 277 . . . . . . . 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 2462 . . . . . . . . . 10 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
71 euex 2056 . . . . . . . . . 10 (∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
7270, 71sylbi 121 . . . . . . . . 9 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
73 eqid 2177 . . . . . . . . . . . . . . 15 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7425, 73cnf 13371 . . . . . . . . . . . . . 14 ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊 (𝑅 ×t 𝑆))
751, 3txuni 13430 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
766, 75eqtrid 2222 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 = (𝑅 ×t 𝑆))
77763adant3 1017 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 = (𝑅 ×t 𝑆))
7877adantr 276 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑍 = (𝑅 ×t 𝑆))
7978feq3d 5350 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (:𝑊𝑍:𝑊 (𝑅 ×t 𝑆)))
8074, 79syl5ibr 156 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊𝑍))
8180anim1d 336 . . . . . . . . . . . 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 162 . . . . . . . . . . 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 109 . . . . . . . . . . 11 (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
8482, 83jca2 308 . . . . . . . . . 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 1880 . . . . . . . . 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 2105 . . . . . . 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 411 . . . . . 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 124 . . . . 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 2255 . . . 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 1898 . . 3 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
9392ex 115 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
9422, 93impbid 129 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 104  wb 105  w3a 978   = wceq 1353  wex 1492  ∃!weu 2026  ∃*wmo 2027  wcel 2148  ∃!wreu 2457  Vcvv 2737   cuni 3807   × cxp 4621  cres 4625  ccom 4627  wf 5208  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Topctop 13162  TopOnctopon 13175   Cn ccn 13352   ×t ctx 13419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-topgen 12657  df-top 13163  df-topon 13176  df-bases 13208  df-cn 13355  df-tx 13420
This theorem is referenced by: (None)
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