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Theorem txcn 13069
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 12810 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
3 txcn.2 . . . . 5 𝑌 = 𝑆
43toptopon 12810 . . . 4 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
5 txcn.5 . . . . . . 7 𝑃 = (1st𝑍)
6 txcn.3 . . . . . . . 8 𝑍 = (𝑋 × 𝑌)
76reseq2i 4888 . . . . . . 7 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
85, 7eqtri 2191 . . . . . 6 𝑃 = (1st ↾ (𝑋 × 𝑌))
9 tx1cn 13063 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
108, 9eqeltrid 2257 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
11 txcn.6 . . . . . . 7 𝑄 = (2nd𝑍)
126reseq2i 4888 . . . . . . 7 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
1311, 12eqtri 2191 . . . . . 6 𝑄 = (2nd ↾ (𝑋 × 𝑌))
14 tx2cn 13064 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2257 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
16 cnco 13015 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → (𝑃𝐹) ∈ (𝑈 Cn 𝑅))
17 cnco 13015 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝑄𝐹) ∈ (𝑈 Cn 𝑆))
1816, 17anim12dan 595 . . . . . 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 1012 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
23 cntop1 12995 . . . . . . . 8 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
2423ad2antrl 487 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑈 ∈ Top)
25 txcn.4 . . . . . . . 8 𝑊 = 𝑈
2625topopn 12800 . . . . . . 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 12998 . . . . . . 7 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → (𝑃𝐹):𝑊𝑋)
2928ad2antrl 487 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑃𝐹):𝑊𝑋)
3025, 3cnf 12998 . . . . . . 7 ((𝑄𝐹) ∈ (𝑈 Cn 𝑆) → (𝑄𝐹):𝑊𝑌)
3130ad2antll 488 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑄𝐹):𝑊𝑌)
328, 13upxp 13066 . . . . . . 7 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
33 feq3 5332 . . . . . . . . . 10 (𝑍 = (𝑋 × 𝑌) → (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌)))
346, 33ax-mp 5 . . . . . . . . 9 (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌))
35343anbi1i 1185 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3635eubii 2028 . . . . . . 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 1233 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
39 euex 2049 . . . . 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 1033 . . . . . . 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 12800 . . . . . . . . . 10 (𝑅 ∈ Top → 𝑋𝑅)
443topopn 12800 . . . . . . . . . 10 (𝑆 ∈ Top → 𝑌𝑆)
45 xpexg 4725 . . . . . . . . . . 11 ((𝑋𝑅𝑌𝑆) → (𝑋 × 𝑌) ∈ V)
466, 45eqeltrid 2257 . . . . . . . . . 10 ((𝑋𝑅𝑌𝑆) → 𝑍 ∈ V)
4743, 44, 46syl2an 287 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 ∈ V)
48473adant3 1012 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 ∈ V)
4948ad2antrr 485 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑍 ∈ V)
50 fex2 5366 . . . . . . 7 ((𝐹:𝑊𝑍𝑊𝑈𝑍 ∈ V) → 𝐹 ∈ V)
5141, 42, 49, 50syl3anc 1233 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹 ∈ V)
52 eumo 2051 . . . . . . . 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 977 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
57 coeq2 4769 . . . . . . . . . . . 12 (𝐹 = → (𝑃𝐹) = (𝑃))
58 coeq2 4769 . . . . . . . . . . . 12 (𝐹 = → (𝑄𝐹) = (𝑄))
5957, 58jca 304 . . . . . . . . . . 11 (𝐹 = → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6059eqcoms 2173 . . . . . . . . . 10 ( = 𝐹 → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6160biantrud 302 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍 ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))))
62 feq1 5330 . . . . . . . . 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 2911 . . . . . 6 (((𝐹 ∈ V ∧ ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ∧ ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ 𝐹:𝑊𝑍)) → = 𝐹)
6651, 54, 55, 41, 65syl22anc 1234 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → = 𝐹)
67 eqid 2170 . . . . . . . . . 10 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6867, 1, 3, 6, 5, 11uptx 13068 . . . . . . . . 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 2455 . . . . . . . . . 10 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
71 euex 2049 . . . . . . . . . 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 2170 . . . . . . . . . . . . . . 15 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7425, 73cnf 12998 . . . . . . . . . . . . . 14 ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊 (𝑅 ×t 𝑆))
751, 3txuni 13057 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
766, 75eqtrid 2215 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 = (𝑅 ×t 𝑆))
77763adant3 1012 . . . . . . . . . . . . . . . 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 5336 . . . . . . . . . . . . . 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 1873 . . . . . . . . 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 2098 . . . . . . 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 2248 . . . 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 1891 . . 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 973   = wceq 1348  wex 1485  ∃!weu 2019  ∃*wmo 2020  wcel 2141  ∃!wreu 2450  Vcvv 2730   cuni 3796   × cxp 4609  cres 4613  ccom 4615  wf 5194  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  Topctop 12789  TopOnctopon 12802   Cn ccn 12979   ×t ctx 13046
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047
This theorem is referenced by: (None)
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