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Theorem txpconn 33826
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 33819 . . 3 (𝑅 ∈ PConn → 𝑅 ∈ Top)
2 pconntop 33819 . . 3 (𝑆 ∈ PConn → 𝑆 ∈ Top)
3 txtop 22920 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 596 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ Top)
5 an6 1445 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)))
6 eqid 2736 . . . . . . . . . . . 12 𝑅 = 𝑅
76pconncn 33818 . . . . . . . . . . 11 ((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8 eqid 2736 . . . . . . . . . . . 12 𝑆 = 𝑆
98pconncn 33818 . . . . . . . . . . 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 234 . . . . . . . . 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 3217 . . . . . . . . 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 233 . . . . . . . 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 24244 . . . . . . . . . . . . 13 (0[,]1) = II
15 eqid 2736 . . . . . . . . . . . . 13 (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)
1614, 15txcnmpt 22975 . . . . . . . . . . . 12 ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
1716ad2antrl 726 . . . . . . . . . . 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 13386 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
19 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑔𝑡) = (𝑔‘0))
20 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡) = (‘0))
2119, 20opeq12d 4838 . . . . . . . . . . . . . 14 (𝑡 = 0 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘0), (‘0)⟩)
22 opex 5421 . . . . . . . . . . . . . 14 ⟨(𝑔‘0), (‘0)⟩ ∈ V
2321, 15, 22fvmpt 6948 . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 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‘0) = 𝑥)
27 simprrr 780 . . . . . . . . . . . . . 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 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)))) → (‘0) = 𝑦)
2926, 28opeq12d 4838 . . . . . . . . . . . 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 2788 . . . . . . . . . . 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 13387 . . . . . . . . . . . . 13 1 ∈ (0[,]1)
32 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑔𝑡) = (𝑔‘1))
33 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑡) = (‘1))
3432, 33opeq12d 4838 . . . . . . . . . . . . . 14 (𝑡 = 1 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘1), (‘1)⟩)
35 opex 5421 . . . . . . . . . . . . . 14 ⟨(𝑔‘1), (‘1)⟩ ∈ V
3634, 15, 35fvmpt 6948 . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 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 4838 . . . . . . . . . . . 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 2788 . . . . . . . . . . 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 6841 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0))
4342eqeq1d 2738 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44 fveq1 6841 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1))
4544eqeq1d 2738 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
4643, 45anbi12d 631 . . . . . . . . . . . 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 3581 . . . . . . . . . . 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 835 . . . . . . . . . 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 457 . . . . . . . . 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 3205 . . . . . . . 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 1118 . . . . . 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 3197 . . . . 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 3197 . . . 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 2748 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5655anbi2d 629 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5756rexbidv 3175 . . . . . . 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 5797 . . . . . 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 2748 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
6059anbi1d 630 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6160rexbidv 3175 . . . . . . 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 3212 . . . . . 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 282 . . . . 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 5797 . . . 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 233 . . 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 22943 . . . . 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 3313 . . . 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 3319 . . 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 231 . 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 2736 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7271ispconn 33817 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  cop 4592   cuni 4865  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052  [,]cicc 13267  Topctop 22242   Cn ccn 22575   ×t ctx 22911  IIcii 24238  PConncpconn 33813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-tx 22913  df-ii 24240  df-pconn 33815
This theorem is referenced by:  txsconn  33835
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