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Theorem txcn 21709
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 21001 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
3 txcn.2 . . . . 5 𝑌 = 𝑆
43toptopon 21001 . . . 4 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
5 txcn.5 . . . . . . 7 𝑃 = (1st𝑍)
6 txcn.3 . . . . . . . 8 𝑍 = (𝑋 × 𝑌)
76reseq2i 5562 . . . . . . 7 (1st𝑍) = (1st ↾ (𝑋 × 𝑌))
85, 7eqtri 2787 . . . . . 6 𝑃 = (1st ↾ (𝑋 × 𝑌))
9 tx1cn 21692 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
108, 9syl5eqel 2848 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
11 txcn.6 . . . . . . 7 𝑄 = (2nd𝑍)
126reseq2i 5562 . . . . . . 7 (2nd𝑍) = (2nd ↾ (𝑋 × 𝑌))
1311, 12eqtri 2787 . . . . . 6 𝑄 = (2nd ↾ (𝑋 × 𝑌))
14 tx2cn 21693 . . . . . 6 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
1513, 14syl5eqel 2848 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
16 cnco 21350 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑃 ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → (𝑃𝐹) ∈ (𝑈 Cn 𝑅))
17 cnco 21350 . . . . . . 7 ((𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ 𝑄 ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → (𝑄𝐹) ∈ (𝑈 Cn 𝑆))
1816, 17anim12dan 612 . . . . . 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 402 . . . . 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 579 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
212, 4, 20syl2anb 591 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
22213adant3 1162 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))
23 cntop1 21324 . . . . . . . 8 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → 𝑈 ∈ Top)
2423ad2antrl 719 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑈 ∈ Top)
25 txcn.4 . . . . . . . 8 𝑊 = 𝑈
2625topopn 20990 . . . . . . 7 (𝑈 ∈ Top → 𝑊𝑈)
2724, 26syl 17 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑊𝑈)
2825, 1cnf 21330 . . . . . . 7 ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) → (𝑃𝐹):𝑊𝑋)
2928ad2antrl 719 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑃𝐹):𝑊𝑋)
3025, 3cnf 21330 . . . . . . 7 ((𝑄𝐹) ∈ (𝑈 Cn 𝑆) → (𝑄𝐹):𝑊𝑌)
3130ad2antll 720 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (𝑄𝐹):𝑊𝑌)
328, 13upxp 21706 . . . . . . 7 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
33 feq3 6206 . . . . . . . . . 10 (𝑍 = (𝑋 × 𝑌) → (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌)))
346, 33ax-mp 5 . . . . . . . . 9 (:𝑊𝑍:𝑊⟶(𝑋 × 𝑌))
35343anbi1i 1196 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3635eubii 2584 . . . . . . 7 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!(:𝑊⟶(𝑋 × 𝑌) ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3732, 36sylibr 225 . . . . . 6 ((𝑊𝑈 ∧ (𝑃𝐹):𝑊𝑋 ∧ (𝑄𝐹):𝑊𝑌) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
3827, 29, 31, 37syl3anc 1490 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
39 euex 2591 . . . . 5 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
4038, 39syl 17 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
41 simpll3 1273 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹:𝑊𝑍)
4227adantr 472 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑊𝑈)
431topopn 20990 . . . . . . . . . 10 (𝑅 ∈ Top → 𝑋𝑅)
443topopn 20990 . . . . . . . . . 10 (𝑆 ∈ Top → 𝑌𝑆)
45 xpexg 7158 . . . . . . . . . . 11 ((𝑋𝑅𝑌𝑆) → (𝑋 × 𝑌) ∈ V)
466, 45syl5eqel 2848 . . . . . . . . . 10 ((𝑋𝑅𝑌𝑆) → 𝑍 ∈ V)
4743, 44, 46syl2an 589 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 ∈ V)
48473adant3 1162 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 ∈ V)
4948ad2antrr 717 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝑍 ∈ V)
50 fex2 7319 . . . . . . 7 ((𝐹:𝑊𝑍𝑊𝑈𝑍 ∈ V) → 𝐹 ∈ V)
5141, 42, 49, 50syl3anc 1490 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → 𝐹 ∈ V)
52 eumo 2592 . . . . . . . 8 (∃!(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
5338, 52syl 17 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
5453adantr 472 . . . . . 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 477 . . . . . 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 1116 . . . . . . . 8 ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
57 coeq2 5449 . . . . . . . . . . . 12 (𝐹 = → (𝑃𝐹) = (𝑃))
58 coeq2 5449 . . . . . . . . . . . 12 (𝐹 = → (𝑄𝐹) = (𝑄))
5957, 58jca 507 . . . . . . . . . . 11 (𝐹 = → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6059eqcoms 2773 . . . . . . . . . 10 ( = 𝐹 → ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6160biantrud 527 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍 ↔ (:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))))
62 feq1 6204 . . . . . . . . 9 ( = 𝐹 → (:𝑊𝑍𝐹:𝑊𝑍))
6361, 62bitr3d 272 . . . . . . . 8 ( = 𝐹 → ((:𝑊𝑍 ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ↔ 𝐹:𝑊𝑍))
6456, 63syl5bb 274 . . . . . . 7 ( = 𝐹 → ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ 𝐹:𝑊𝑍))
6564moi2 3546 . . . . . 6 (((𝐹 ∈ V ∧ ∃*(:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) ∧ ((:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ∧ 𝐹:𝑊𝑍)) → = 𝐹)
6651, 54, 55, 41, 65syl22anc 867 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) ∧ (:𝑊𝑍 ∧ (𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → = 𝐹)
67 eqid 2765 . . . . . . . . . 10 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
6867, 1, 3, 6, 5, 11uptx 21708 . . . . . . . . 9 (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)))
6968adantl 473 . . . . . . . 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 3062 . . . . . . . . . 10 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) ↔ ∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
71 euex 2591 . . . . . . . . . 10 (∃!( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
7270, 71sylbi 208 . . . . . . . . 9 (∃! ∈ (𝑈 Cn (𝑅 ×t 𝑆))((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄)) → ∃( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))))
73 eqid 2765 . . . . . . . . . . . . . . 15 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7425, 73cnf 21330 . . . . . . . . . . . . . 14 ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊 (𝑅 ×t 𝑆))
751, 3txuni 21675 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
766, 75syl5eq 2811 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝑍 = (𝑅 ×t 𝑆))
77763adant3 1162 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → 𝑍 = (𝑅 ×t 𝑆))
7877adantr 472 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝑍 = (𝑅 ×t 𝑆))
7978feq3d 6210 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (:𝑊𝑍:𝑊 (𝑅 ×t 𝑆)))
8074, 79syl5ibr 237 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → ( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) → :𝑊𝑍))
8180anim1d 604 . . . . . . . . . . . 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 243 . . . . . . . . . . 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 474 . . . . . . . . . . . 12 (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
8483a1i 11 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → (( ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ∧ ((𝑃𝐹) = (𝑃) ∧ (𝑄𝐹) = (𝑄))) → ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
8582, 84jcad 508 . . . . . . . . . 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)))))
8685eximdv 2012 . . . . . . . . 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)))))
8772, 86syl5 34 . . . . . . . 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)))))
8869, 87mpd 15 . . . . . . 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))))
89 eupick 2658 . . . . . . 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))))
9038, 88, 89syl2anc 579 . . . . . 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))))
9190imp 395 . . . . 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)))
9266, 91eqeltrrd 2845 . . . 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)))
9340, 92exlimddv 2030 . . 3 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) ∧ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))
9493ex 401 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆)) → 𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆))))
9522, 94impbid 203 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 setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  ∃*wmo 2563  ∃!weu 2581  ∃!wreu 3057  Vcvv 3350   cuni 4594   × cxp 5275  cres 5279  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Topctop 20977  TopOnctopon 20994   Cn ccn 21308   ×t ctx 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-topgen 16370  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-tx 21645
This theorem is referenced by: (None)
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