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Theorem txcn 23523
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 = βˆͺ 𝑅
txcn.2 π‘Œ = βˆͺ 𝑆
txcn.3 𝑍 = (𝑋 Γ— π‘Œ)
txcn.4 π‘Š = βˆͺ π‘ˆ
txcn.5 𝑃 = (1st β†Ύ 𝑍)
txcn.6 𝑄 = (2nd β†Ύ 𝑍)
Assertion
Ref Expression
txcn ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
Proof of Theorem txcn
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 txcn.1 . . . . 5 𝑋 = βˆͺ 𝑅
21toptopon 22812 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOnβ€˜π‘‹))
3 txcn.2 . . . . 5 π‘Œ = βˆͺ 𝑆
43toptopon 22812 . . . 4 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOnβ€˜π‘Œ))
5 txcn.5 . . . . . . 7 𝑃 = (1st β†Ύ 𝑍)
6 txcn.3 . . . . . . . 8 𝑍 = (𝑋 Γ— π‘Œ)
76reseq2i 5976 . . . . . . 7 (1st β†Ύ 𝑍) = (1st β†Ύ (𝑋 Γ— π‘Œ))
85, 7eqtri 2756 . . . . . 6 𝑃 = (1st β†Ύ (𝑋 Γ— π‘Œ))
9 tx1cn 23506 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
108, 9eqeltrid 2833 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑃 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
11 txcn.6 . . . . . . 7 𝑄 = (2nd β†Ύ 𝑍)
126reseq2i 5976 . . . . . . 7 (2nd β†Ύ 𝑍) = (2nd β†Ύ (𝑋 Γ— π‘Œ))
1311, 12eqtri 2756 . . . . . 6 𝑄 = (2nd β†Ύ (𝑋 Γ— π‘Œ))
14 tx2cn 23507 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2833 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
16 cnco 23163 . . . . . . 7 ((𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ 𝑃 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅)) β†’ (𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅))
17 cnco 23163 . . . . . . 7 ((𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆)) β†’ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))
1816, 17anim12dan 618 . . . . . 6 ((𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ (𝑃 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))) β†’ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆)))
1918expcom 413 . . . . 5 ((𝑃 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅) ∧ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆)) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
2010, 15, 19syl2anc 583 . . . 4 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
212, 4, 20syl2anb 597 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
22213adant3 1130 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
23 cntop1 23137 . . . . . . . 8 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ π‘ˆ ∈ Top)
2423ad2antrl 727 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ π‘ˆ ∈ Top)
25 txcn.4 . . . . . . . 8 π‘Š = βˆͺ π‘ˆ
2625topopn 22801 . . . . . . 7 (π‘ˆ ∈ Top β†’ π‘Š ∈ π‘ˆ)
2724, 26syl 17 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ π‘Š ∈ π‘ˆ)
2825, 1cnf 23143 . . . . . . 7 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹)
2928ad2antrl 727 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹)
3025, 3cnf 23143 . . . . . . 7 ((𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆) β†’ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ)
3130ad2antll 728 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ)
328, 13upxp 23520 . . . . . . 7 ((π‘Š ∈ π‘ˆ ∧ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹ ∧ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ) β†’ βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆ(𝑋 Γ— π‘Œ) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
33 feq3 6699 . . . . . . . . . 10 (𝑍 = (𝑋 Γ— π‘Œ) β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ β„Ž:π‘ŠβŸΆ(𝑋 Γ— π‘Œ)))
346, 33ax-mp 5 . . . . . . . . 9 (β„Ž:π‘ŠβŸΆπ‘ ↔ β„Ž:π‘ŠβŸΆ(𝑋 Γ— π‘Œ))
35343anbi1i 1155 . . . . . . . 8 ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ (β„Ž:π‘ŠβŸΆ(𝑋 Γ— π‘Œ) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
3635eubii 2575 . . . . . . 7 (βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆ(𝑋 Γ— π‘Œ) ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
3732, 36sylibr 233 . . . . . 6 ((π‘Š ∈ π‘ˆ ∧ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹ ∧ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ) β†’ βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
3827, 29, 31, 37syl3anc 1369 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
39 euex 2567 . . . . 5 (βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ βˆƒβ„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
4038, 39syl 17 . . . 4 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ βˆƒβ„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
41 simpll3 1212 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ 𝐹:π‘ŠβŸΆπ‘)
4227adantr 480 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ π‘Š ∈ π‘ˆ)
4341, 42fexd 7233 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ 𝐹 ∈ V)
44 eumo 2568 . . . . . . . 8 (βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ βˆƒ*β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
4538, 44syl 17 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ βˆƒ*β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
4645adantr 480 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ βˆƒ*β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
47 simpr 484 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
48 3anass 1093 . . . . . . . 8 ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ (β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))))
49 coeq2 5855 . . . . . . . . . . . 12 (𝐹 = β„Ž β†’ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž))
50 coeq2 5855 . . . . . . . . . . . 12 (𝐹 = β„Ž β†’ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))
5149, 50jca 511 . . . . . . . . . . 11 (𝐹 = β„Ž β†’ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
5251eqcoms 2736 . . . . . . . . . 10 (β„Ž = 𝐹 β†’ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
5352biantrud 531 . . . . . . . . 9 (β„Ž = 𝐹 β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ (β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))))
54 feq1 6697 . . . . . . . . 9 (β„Ž = 𝐹 β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ 𝐹:π‘ŠβŸΆπ‘))
5553, 54bitr3d 281 . . . . . . . 8 (β„Ž = 𝐹 β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) ↔ 𝐹:π‘ŠβŸΆπ‘))
5648, 55bitrid 283 . . . . . . 7 (β„Ž = 𝐹 β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ 𝐹:π‘ŠβŸΆπ‘))
5756moi2 3710 . . . . . 6 (((𝐹 ∈ V ∧ βˆƒ*β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) ∧ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ 𝐹:π‘ŠβŸΆπ‘)) β†’ β„Ž = 𝐹)
5843, 46, 47, 41, 57syl22anc 838 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ β„Ž = 𝐹)
59 eqid 2728 . . . . . . . . . 10 (𝑅 Γ—t 𝑆) = (𝑅 Γ—t 𝑆)
6059, 1, 3, 6, 5, 11uptx 23522 . . . . . . . . 9 (((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆)) β†’ βˆƒ!β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
6160adantl 481 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ βˆƒ!β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
62 df-reu 3373 . . . . . . . . . 10 (βˆƒ!β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ βˆƒ!β„Ž(β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))))
63 euex 2567 . . . . . . . . . 10 (βˆƒ!β„Ž(β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ βˆƒβ„Ž(β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))))
6462, 63sylbi 216 . . . . . . . . 9 (βˆƒ!β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ βˆƒβ„Ž(β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))))
65 eqid 2728 . . . . . . . . . . . . . . 15 βˆͺ (𝑅 Γ—t 𝑆) = βˆͺ (𝑅 Γ—t 𝑆)
6625, 65cnf 23143 . . . . . . . . . . . . . 14 (β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ β„Ž:π‘ŠβŸΆβˆͺ (𝑅 Γ—t 𝑆))
671, 3txuni 23489 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝑅 Γ—t 𝑆))
686, 67eqtrid 2780 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
69683adant3 1130 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
7069adantr 480 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
7170feq3d 6703 . . . . . . . . . . . . . 14 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ β„Ž:π‘ŠβŸΆβˆͺ (𝑅 Γ—t 𝑆)))
7266, 71imbitrrid 245 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ β„Ž:π‘ŠβŸΆπ‘))
7372anim1d 610 . . . . . . . . . . . 12 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ ((β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ (β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))))
7473, 48imbitrrdi 251 . . . . . . . . . . 11 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ ((β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))))
75 simpl 482 . . . . . . . . . . 11 ((β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))
7674, 75jca2 513 . . . . . . . . . 10 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ ((β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))))
7776eximdv 1913 . . . . . . . . 9 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (βˆƒβ„Ž(β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ βˆƒβ„Ž((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))))
7864, 77syl5 34 . . . . . . . 8 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ (βˆƒ!β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ βˆƒβ„Ž((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))))
7961, 78mpd 15 . . . . . . 7 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ βˆƒβ„Ž((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))))
80 eupick 2625 . . . . . . 7 ((βˆƒ!β„Ž(β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ βˆƒβ„Ž((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ∧ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))) β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))))
8138, 79, 80syl2anc 583 . . . . . 6 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) β†’ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))))
8281imp 406 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))
8358, 82eqeltrrd 2830 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) ∧ (β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) β†’ 𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))
8440, 83exlimddv 1931 . . 3 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) ∧ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))) β†’ 𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)))
8584ex 412 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ (((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆)) β†’ 𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆))))
8622, 85impbid 211 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ (𝐹 ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) ↔ ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) ∧ (𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  βˆƒ*wmo 2528  βˆƒ!weu 2558  βˆƒ!wreu 3370  Vcvv 3470  βˆͺ cuni 4903   Γ— cxp 5670   β†Ύ cres 5674   ∘ ccom 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  Topctop 22788  TopOnctopon 22805   Cn ccn 23121   Γ—t ctx 23457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8840  df-topgen 17418  df-top 22789  df-topon 22806  df-bases 22842  df-cn 23124  df-tx 23459
This theorem is referenced by: (None)
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