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Theorem txcn 23474
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 22763 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOnβ€˜π‘‹))
3 txcn.2 . . . . 5 π‘Œ = βˆͺ 𝑆
43toptopon 22763 . . . 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 5969 . . . . . . 7 (1st β†Ύ 𝑍) = (1st β†Ύ (𝑋 Γ— π‘Œ))
85, 7eqtri 2752 . . . . . 6 𝑃 = (1st β†Ύ (𝑋 Γ— π‘Œ))
9 tx1cn 23457 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
108, 9eqeltrid 2829 . . . . 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 5969 . . . . . . 7 (2nd β†Ύ 𝑍) = (2nd β†Ύ (𝑋 Γ— π‘Œ))
1311, 12eqtri 2752 . . . . . 6 𝑄 = (2nd β†Ύ (𝑋 Γ— π‘Œ))
14 tx2cn 23458 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2829 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
16 cnco 23114 . . . . . . 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 23114 . . . . . . 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 1129 . 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 23088 . . . . . . . 8 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ π‘ˆ ∈ Top)
2423ad2antrl 725 . . . . . . 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 22752 . . . . . . 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 23094 . . . . . . 7 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹)
2928ad2antrl 725 . . . . . 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 23094 . . . . . . 7 ((𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆) β†’ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ)
3130ad2antll 726 . . . . . 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 23471 . . . . . . 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 6691 . . . . . . . . . 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 1154 . . . . . . . 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 2571 . . . . . . 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 1368 . . . . 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 2563 . . . . 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 1211 . . . . . . 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 7221 . . . . . 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 2564 . . . . . . . 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 1092 . . . . . . . 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 5849 . . . . . . . . . . . 12 (𝐹 = β„Ž β†’ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž))
50 coeq2 5849 . . . . . . . . . . . 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 2732 . . . . . . . . . 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 6689 . . . . . . . . 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 3705 . . . . . 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 836 . . . . 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 2724 . . . . . . . . . 10 (𝑅 Γ—t 𝑆) = (𝑅 Γ—t 𝑆)
6059, 1, 3, 6, 5, 11uptx 23473 . . . . . . . . 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 3369 . . . . . . . . . 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 2563 . . . . . . . . . 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 2724 . . . . . . . . . . . . . . 15 βˆͺ (𝑅 Γ—t 𝑆) = βˆͺ (𝑅 Γ—t 𝑆)
6625, 65cnf 23094 . . . . . . . . . . . . . 14 (β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ β„Ž:π‘ŠβŸΆβˆͺ (𝑅 Γ—t 𝑆))
671, 3txuni 23440 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝑅 Γ—t 𝑆))
686, 67eqtrid 2776 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
69683adant3 1129 . . . . . . . . . . . . . . . 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 6695 . . . . . . . . . . . . . 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 1912 . . . . . . . . 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 2621 . . . . . . 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 2826 . . . 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 1930 . . 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 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆƒ*wmo 2524  βˆƒ!weu 2554  βˆƒ!wreu 3366  Vcvv 3466  βˆͺ cuni 4900   Γ— cxp 5665   β†Ύ cres 5669   ∘ ccom 5671  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  Topctop 22739  TopOnctopon 22756   Cn ccn 23072   Γ—t ctx 23408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cn 23075  df-tx 23410
This theorem is referenced by: (None)
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