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Theorem txcn 13778
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 13521 . . . 4 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOnβ€˜π‘‹))
3 txcn.2 . . . . 5 π‘Œ = βˆͺ 𝑆
43toptopon 13521 . . . 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 4905 . . . . . . 7 (1st β†Ύ 𝑍) = (1st β†Ύ (𝑋 Γ— π‘Œ))
85, 7eqtri 2198 . . . . . 6 𝑃 = (1st β†Ύ (𝑋 Γ— π‘Œ))
9 tx1cn 13772 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑅))
108, 9eqeltrid 2264 . . . . 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 4905 . . . . . . 7 (2nd β†Ύ 𝑍) = (2nd β†Ύ (𝑋 Γ— π‘Œ))
1311, 12eqtri 2198 . . . . . 6 𝑄 = (2nd β†Ύ (𝑋 Γ— π‘Œ))
14 tx2cn 13773 . . . . . 6 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ (2nd β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
1513, 14eqeltrid 2264 . . . . 5 ((𝑅 ∈ (TopOnβ€˜π‘‹) ∧ 𝑆 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝑄 ∈ ((𝑅 Γ—t 𝑆) Cn 𝑆))
16 cnco 13724 . . . . . . 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 13724 . . . . . . 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 600 . . . . . 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 116 . . . . 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 411 . . . 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 291 . . 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 1017 . 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 13704 . . . . . . . 8 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ π‘ˆ ∈ Top)
2423ad2antrl 490 . . . . . . 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 13511 . . . . . . 7 (π‘ˆ ∈ Top β†’ π‘Š ∈ π‘ˆ)
2724, 26syl 14 . . . . . 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 13707 . . . . . . 7 ((𝑃 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑅) β†’ (𝑃 ∘ 𝐹):π‘ŠβŸΆπ‘‹)
2928ad2antrl 490 . . . . . 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 13707 . . . . . . 7 ((𝑄 ∘ 𝐹) ∈ (π‘ˆ Cn 𝑆) β†’ (𝑄 ∘ 𝐹):π‘ŠβŸΆπ‘Œ)
3130ad2antll 491 . . . . . 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 13775 . . . . . . 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 5351 . . . . . . . . . 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 1190 . . . . . . . 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 2035 . . . . . . 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 134 . . . . . 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 1238 . . . . 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 2056 . . . . 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 14 . . . 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 1038 . . . . . . 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 276 . . . . . . 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)
431topopn 13511 . . . . . . . . . 10 (𝑅 ∈ Top β†’ 𝑋 ∈ 𝑅)
443topopn 13511 . . . . . . . . . 10 (𝑆 ∈ Top β†’ π‘Œ ∈ 𝑆)
45 xpexg 4741 . . . . . . . . . . 11 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 Γ— π‘Œ) ∈ V)
466, 45eqeltrid 2264 . . . . . . . . . 10 ((𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑆) β†’ 𝑍 ∈ V)
4743, 44, 46syl2an 289 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ 𝑍 ∈ V)
48473adant3 1017 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ 𝑍 ∈ V)
4948ad2antrr 488 . . . . . . 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†’ 𝑍 ∈ V)
50 fex2 5385 . . . . . . 7 ((𝐹:π‘ŠβŸΆπ‘ ∧ π‘Š ∈ π‘ˆ ∧ 𝑍 ∈ V) β†’ 𝐹 ∈ V)
5141, 42, 49, 50syl3anc 1238 . . . . . 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)
52 eumo 2058 . . . . . . . 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)))
5338, 52syl 14 . . . . . . 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)))
5453adantr 276 . . . . . 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)))
55 simpr 110 . . . . . 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)))
56 3anass 982 . . . . . . . 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))))
57 coeq2 4786 . . . . . . . . . . . 12 (𝐹 = β„Ž β†’ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž))
58 coeq2 4786 . . . . . . . . . . . 12 (𝐹 = β„Ž β†’ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))
5957, 58jca 306 . . . . . . . . . . 11 (𝐹 = β„Ž β†’ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
6059eqcoms 2180 . . . . . . . . . 10 (β„Ž = 𝐹 β†’ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))
6160biantrud 304 . . . . . . . . 9 (β„Ž = 𝐹 β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ (β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)))))
62 feq1 5349 . . . . . . . . 9 (β„Ž = 𝐹 β†’ (β„Ž:π‘ŠβŸΆπ‘ ↔ 𝐹:π‘ŠβŸΆπ‘))
6361, 62bitr3d 190 . . . . . . . 8 (β„Ž = 𝐹 β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ ((𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž))) ↔ 𝐹:π‘ŠβŸΆπ‘))
6456, 63bitrid 192 . . . . . . 7 (β„Ž = 𝐹 β†’ ((β„Ž:π‘ŠβŸΆπ‘ ∧ (𝑃 ∘ 𝐹) = (𝑃 ∘ β„Ž) ∧ (𝑄 ∘ 𝐹) = (𝑄 ∘ β„Ž)) ↔ 𝐹:π‘ŠβŸΆπ‘))
6564moi2 2919 . . . . . 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)
6651, 54, 55, 41, 65syl22anc 1239 . . . . 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)
67 eqid 2177 . . . . . . . . . 10 (𝑅 Γ—t 𝑆) = (𝑅 Γ—t 𝑆)
6867, 1, 3, 6, 5, 11uptx 13777 . . . . . . . . 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)))
6968adantl 277 . . . . . . . 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)))
70 df-reu 2462 . . . . . . . . . 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))))
71 euex 2056 . . . . . . . . . 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))))
7270, 71sylbi 121 . . . . . . . . 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))))
73 eqid 2177 . . . . . . . . . . . . . . 15 βˆͺ (𝑅 Γ—t 𝑆) = βˆͺ (𝑅 Γ—t 𝑆)
7425, 73cnf 13707 . . . . . . . . . . . . . 14 (β„Ž ∈ (π‘ˆ Cn (𝑅 Γ—t 𝑆)) β†’ β„Ž:π‘ŠβŸΆβˆͺ (𝑅 Γ—t 𝑆))
751, 3txuni 13766 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ (𝑋 Γ— π‘Œ) = βˆͺ (𝑅 Γ—t 𝑆))
766, 75eqtrid 2222 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
77763adant3 1017 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:π‘ŠβŸΆπ‘) β†’ 𝑍 = βˆͺ (𝑅 Γ—t 𝑆))
7877adantr 276 . . . . . . . . . . . . . . 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))
7978feq3d 5355 . . . . . . . . . . . . . 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)))
8074, 79imbitrrid 156 . . . . . . . . . . . . 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‘))
8180anim1d 336 . . . . . . . . . . . 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)))))
8281, 56imbitrrdi 162 . . . . . . . . . . 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))))
83 simpl 109 . . . . . . . . . . 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)))
8482, 83jca2 308 . . . . . . . . . 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)))))
8584eximdv 1880 . . . . . . . . 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)))))
8672, 85syl5 32 . . . . . . . 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)))))
8769, 86mpd 13 . . . . . . 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))))
88 eupick 2105 . . . . . . 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))))
8938, 87, 88syl2anc 411 . . . . . 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))))
9089imp 124 . . . . 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)))
9166, 90eqeltrrd 2255 . . . 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)))
9240, 91exlimddv 1898 . . 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)))
9392ex 115 . 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))))
9422, 93impbid 129 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 set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  βˆƒwex 1492  βˆƒ!weu 2026  βˆƒ*wmo 2027   ∈ wcel 2148  βˆƒ!wreu 2457  Vcvv 2738  βˆͺ cuni 3810   Γ— cxp 4625   β†Ύ cres 4629   ∘ ccom 4631  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875  1st c1st 6139  2nd c2nd 6140  Topctop 13500  TopOnctopon 13513   Cn ccn 13688   Γ—t ctx 13755
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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-topgen 12709  df-top 13501  df-topon 13514  df-bases 13546  df-cn 13691  df-tx 13756
This theorem is referenced by: (None)
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