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Theorem txpconn 35237
Description: The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
Assertion
Ref Expression
txpconn ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)

Proof of Theorem txpconn
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pconntop 35230 . . 3 (𝑅 ∈ PConn → 𝑅 ∈ Top)
2 pconntop 35230 . . 3 (𝑆 ∈ PConn → 𝑆 ∈ Top)
3 txtop 23577 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 596 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ Top)
5 an6 1447 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)))
6 eqid 2737 . . . . . . . . . . . 12 𝑅 = 𝑅
76pconncn 35229 . . . . . . . . . . 11 ((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8 eqid 2737 . . . . . . . . . . . 12 𝑆 = 𝑆
98pconncn 35229 . . . . . . . . . . 11 ((𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆) → ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤))
107, 9anim12i 613 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
115, 10sylbir 235 . . . . . . . . 9 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
12 reeanv 3229 . . . . . . . . 9 (∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) ↔ (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
1311, 12sylibr 234 . . . . . . . 8 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
14 iiuni 24907 . . . . . . . . . . . . 13 (0[,]1) = II
15 eqid 2737 . . . . . . . . . . . . 13 (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)
1614, 15txcnmpt 23632 . . . . . . . . . . . 12 ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
1716ad2antrl 728 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
18 0elunit 13509 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
19 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑔𝑡) = (𝑔‘0))
20 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡) = (‘0))
2119, 20opeq12d 4881 . . . . . . . . . . . . . 14 (𝑡 = 0 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘0), (‘0)⟩)
22 opex 5469 . . . . . . . . . . . . . 14 ⟨(𝑔‘0), (‘0)⟩ ∈ V
2321, 15, 22fvmpt 7016 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨(𝑔‘0), (‘0)⟩)
2418, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨(𝑔‘0), (‘0)⟩
25 simprrl 781 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
2625simpld 494 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑔‘0) = 𝑥)
27 simprrr 782 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((‘0) = 𝑦 ∧ (‘1) = 𝑤))
2827simpld 494 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (‘0) = 𝑦)
2926, 28opeq12d 4881 . . . . . . . . . . . 12 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ⟨(𝑔‘0), (‘0)⟩ = ⟨𝑥, 𝑦⟩)
3024, 29eqtrid 2789 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩)
31 1elunit 13510 . . . . . . . . . . . . 13 1 ∈ (0[,]1)
32 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑔𝑡) = (𝑔‘1))
33 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑡) = (‘1))
3432, 33opeq12d 4881 . . . . . . . . . . . . . 14 (𝑡 = 1 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘1), (‘1)⟩)
35 opex 5469 . . . . . . . . . . . . . 14 ⟨(𝑔‘1), (‘1)⟩ ∈ V
3634, 15, 35fvmpt 7016 . . . . . . . . . . . . 13 (1 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨(𝑔‘1), (‘1)⟩)
3731, 36ax-mp 5 . . . . . . . . . . . 12 ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨(𝑔‘1), (‘1)⟩
3825simprd 495 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑔‘1) = 𝑧)
3927simprd 495 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (‘1) = 𝑤)
4038, 39opeq12d 4881 . . . . . . . . . . . 12 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ⟨(𝑔‘1), (‘1)⟩ = ⟨𝑧, 𝑤⟩)
4137, 40eqtrid 2789 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)
42 fveq1 6905 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0))
4342eqeq1d 2739 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44 fveq1 6905 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1))
4544eqeq1d 2739 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
4643, 45anbi12d 632 . . . . . . . . . . . 12 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)))
4746rspcev 3622 . . . . . . . . . . 11 (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
4817, 30, 41, 47syl12anc 837 . . . . . . . . . 10 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
4948expr 456 . . . . . . . . 9 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ (𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆))) → ((((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5049rexlimdvva 3213 . . . . . . . 8 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5113, 50mpd 15 . . . . . . 7 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
52513expa 1119 . . . . . 6 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆)) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5352ralrimivva 3202 . . . . 5 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆)) → ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5453ralrimivva 3202 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑥 𝑅𝑦 𝑆𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55 eqeq2 2749 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5655anbi2d 630 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5756rexbidv 3179 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5857ralxp 5852 . . . . . 6 (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
59 eqeq2 2749 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
6059anbi1d 631 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6160rexbidv 3179 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
62612ralbidv 3221 . . . . . 6 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6358, 62bitrid 283 . . . . 5 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6463ralxp 5852 . . . 4 (∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑥 𝑅𝑦 𝑆𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
6554, 64sylibr 234 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
666, 8txuni 23600 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
671, 2, 66syl2an 596 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6867raleqdv 3326 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
6967, 68raleqbidv 3346 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
7065, 69mpbid 232 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71 eqid 2737 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7271ispconn 35228 . 2 ((𝑅 ×t 𝑆) ∈ PConn ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
734, 70, 72sylanbrc 583 1 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cop 4632   cuni 4907  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  [,]cicc 13390  Topctop 22899   Cn ccn 23232   ×t ctx 23568  IIcii 24901  PConncpconn 35224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-tx 23570  df-ii 24903  df-pconn 35226
This theorem is referenced by:  txsconn  35246
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