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Theorem txpconn 34689
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 34682 . . 3 (𝑅 ∈ PConn → 𝑅 ∈ Top)
2 pconntop 34682 . . 3 (𝑆 ∈ PConn → 𝑆 ∈ Top)
3 txtop 23394 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 595 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ Top)
5 an6 1444 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)))
6 eqid 2731 . . . . . . . . . . . 12 𝑅 = 𝑅
76pconncn 34681 . . . . . . . . . . 11 ((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8 eqid 2731 . . . . . . . . . . . 12 𝑆 = 𝑆
98pconncn 34681 . . . . . . . . . . 11 ((𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆) → ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤))
107, 9anim12i 612 . . . . . . . . . 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 3225 . . . . . . . . 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 24722 . . . . . . . . . . . . 13 (0[,]1) = II
15 eqid 2731 . . . . . . . . . . . . 13 (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)
1614, 15txcnmpt 23449 . . . . . . . . . . . 12 ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
1716ad2antrl 725 . . . . . . . . . . 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 13453 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
19 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑔𝑡) = (𝑔‘0))
20 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡) = (‘0))
2119, 20opeq12d 4881 . . . . . . . . . . . . . 14 (𝑡 = 0 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘0), (‘0)⟩)
22 opex 5464 . . . . . . . . . . . . . 14 ⟨(𝑔‘0), (‘0)⟩ ∈ V
2321, 15, 22fvmpt 6998 . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 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)))) → ((‘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 2783 . . . . . . . . . . 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 13454 . . . . . . . . . . . . 13 1 ∈ (0[,]1)
32 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑔𝑡) = (𝑔‘1))
33 fveq2 6891 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑡) = (‘1))
3432, 33opeq12d 4881 . . . . . . . . . . . . . 14 (𝑡 = 1 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘1), (‘1)⟩)
35 opex 5464 . . . . . . . . . . . . . 14 ⟨(𝑔‘1), (‘1)⟩ ∈ V
3634, 15, 35fvmpt 6998 . . . . . . . . . . . . 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 2783 . . . . . . . . . . 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 6890 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0))
4342eqeq1d 2733 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44 fveq1 6890 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1))
4544eqeq1d 2733 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
4643, 45anbi12d 630 . . . . . . . . . . . 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 3612 . . . . . . . . . . 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 834 . . . . . . . . . 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 3210 . . . . . . . 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 1117 . . . . . 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 3199 . . . . 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 3199 . . . 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 2743 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5655anbi2d 628 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5756rexbidv 3177 . . . . . . 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 5841 . . . . . 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 2743 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
6059anbi1d 629 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6160rexbidv 3177 . . . . . . 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 3217 . . . . . 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 5841 . . . 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 23417 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
671, 2, 66syl2an 595 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6867raleqdv 3324 . . . 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 3341 . . 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 2731 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7271ispconn 34680 . 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 582 1 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  cop 4634   cuni 4908  cmpt 5231   × cxp 5674  cfv 6543  (class class class)co 7412  0cc0 11116  1c1 11117  [,]cicc 13334  Topctop 22716   Cn ccn 23049   ×t ctx 23385  IIcii 24716  PConncpconn 34676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-xneg 13099  df-xadd 13100  df-xmul 13101  df-icc 13338  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-topgen 17396  df-psmet 21226  df-xmet 21227  df-met 21228  df-bl 21229  df-mopn 21230  df-top 22717  df-topon 22734  df-bases 22770  df-cn 23052  df-tx 23387  df-ii 24718  df-pconn 34678
This theorem is referenced by:  txsconn  34698
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