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Theorem cnpconn 34749
Description: An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypothesis
Ref Expression
cnpconn.2 𝑌 = 𝐾
Assertion
Ref Expression
cnpconn ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)

Proof of Theorem cnpconn
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntop2 23100 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1132 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 eqid 2726 . . . . . . . . 9 𝐽 = 𝐽
43pconncn 34743 . . . . . . . 8 ((𝐽 ∈ PConn ∧ 𝑢 𝐽𝑣 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
543expb 1117 . . . . . . 7 ((𝐽 ∈ PConn ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
653ad2antl1 1182 . . . . . 6 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7 simprl 768 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔 ∈ (II Cn 𝐽))
8 simpll3 1211 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝐹 ∈ (𝐽 Cn 𝐾))
9 cnco 23125 . . . . . . . 8 ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹𝑔) ∈ (II Cn 𝐾))
107, 8, 9syl2anc 583 . . . . . . 7 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹𝑔) ∈ (II Cn 𝐾))
11 iiuni 24756 . . . . . . . . . . 11 (0[,]1) = II
1211, 3cnf 23105 . . . . . . . . . 10 (𝑔 ∈ (II Cn 𝐽) → 𝑔:(0[,]1)⟶ 𝐽)
137, 12syl 17 . . . . . . . . 9 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → 𝑔:(0[,]1)⟶ 𝐽)
14 0elunit 13452 . . . . . . . . 9 0 ∈ (0[,]1)
15 fvco3 6984 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹𝑔)‘0) = (𝐹‘(𝑔‘0)))
1613, 14, 15sylancl 585 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹𝑔)‘0) = (𝐹‘(𝑔‘0)))
17 simprrl 778 . . . . . . . . 9 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘0) = 𝑢)
1817fveq2d 6889 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹‘(𝑔‘0)) = (𝐹𝑢))
1916, 18eqtrd 2766 . . . . . . 7 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹𝑔)‘0) = (𝐹𝑢))
20 1elunit 13453 . . . . . . . . 9 1 ∈ (0[,]1)
21 fvco3 6984 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑔)‘1) = (𝐹‘(𝑔‘1)))
2213, 20, 21sylancl 585 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹𝑔)‘1) = (𝐹‘(𝑔‘1)))
23 simprrr 779 . . . . . . . . 9 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝑔‘1) = 𝑣)
2423fveq2d 6889 . . . . . . . 8 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → (𝐹‘(𝑔‘1)) = (𝐹𝑣))
2522, 24eqtrd 2766 . . . . . . 7 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ((𝐹𝑔)‘1) = (𝐹𝑣))
26 fveq1 6884 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘0) = ((𝐹𝑔)‘0))
2726eqeq1d 2728 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘0) = (𝐹𝑢) ↔ ((𝐹𝑔)‘0) = (𝐹𝑢)))
28 fveq1 6884 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘1) = ((𝐹𝑔)‘1))
2928eqeq1d 2728 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘1) = (𝐹𝑣) ↔ ((𝐹𝑔)‘1) = (𝐹𝑣)))
3027, 29anbi12d 630 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ (((𝐹𝑔)‘0) = (𝐹𝑢) ∧ ((𝐹𝑔)‘1) = (𝐹𝑣))))
3130rspcev 3606 . . . . . . 7 (((𝐹𝑔) ∈ (II Cn 𝐾) ∧ (((𝐹𝑔)‘0) = (𝐹𝑢) ∧ ((𝐹𝑔)‘1) = (𝐹𝑣))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
3210, 19, 25, 31syl12anc 834 . . . . . 6 ((((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) ∧ (𝑔 ∈ (II Cn 𝐽) ∧ ((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
336, 32rexlimddv 3155 . . . . 5 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
3433ralrimivva 3194 . . . 4 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 𝐽𝑣 𝐽𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
35 cnpconn.2 . . . . . . . . 9 𝑌 = 𝐾
363, 35cnf 23105 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
37363ad2ant3 1132 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
38 forn 6802 . . . . . . . 8 (𝐹:𝑋onto𝑌 → ran 𝐹 = 𝑌)
39383ad2ant2 1131 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40 dffo2 6803 . . . . . . 7 (𝐹: 𝐽onto𝑌 ↔ (𝐹: 𝐽𝑌 ∧ ran 𝐹 = 𝑌))
4137, 39, 40sylanbrc 582 . . . . . 6 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto𝑌)
42 eqeq2 2738 . . . . . . . . 9 ((𝐹𝑣) = 𝑦 → ((𝑓‘1) = (𝐹𝑣) ↔ (𝑓‘1) = 𝑦))
4342anbi2d 628 . . . . . . . 8 ((𝐹𝑣) = 𝑦 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4443rexbidv 3172 . . . . . . 7 ((𝐹𝑣) = 𝑦 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4544cbvfo 7283 . . . . . 6 (𝐹: 𝐽onto𝑌 → (∀𝑣 𝐽𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4641, 45syl 17 . . . . 5 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑣 𝐽𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4746ralbidv 3171 . . . 4 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑢 𝐽𝑣 𝐽𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4834, 47mpbid 231 . . 3 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦))
49 eqeq2 2738 . . . . . . . 8 ((𝐹𝑢) = 𝑥 → ((𝑓‘0) = (𝐹𝑢) ↔ (𝑓‘0) = 𝑥))
5049anbi1d 629 . . . . . . 7 ((𝐹𝑢) = 𝑥 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5150rexbidv 3172 . . . . . 6 ((𝐹𝑢) = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5251ralbidv 3171 . . . . 5 ((𝐹𝑢) = 𝑥 → (∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5352cbvfo 7283 . . . 4 (𝐹: 𝐽onto𝑌 → (∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5441, 53syl 17 . . 3 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → (∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5548, 54mpbid 231 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
5635ispconn 34742 . 2 (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
572, 55, 56sylanbrc 582 1 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wrex 3064   cuni 4902  ran crn 5670  ccom 5673  wf 6533  ontowfo 6535  cfv 6537  (class class class)co 7405  0cc0 11112  1c1 11113  [,]cicc 13333  Topctop 22750   Cn ccn 23083  IIcii 24750  PConncpconn 34738
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 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-icc 13337  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-topgen 17398  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-top 22751  df-topon 22768  df-bases 22804  df-cn 23086  df-ii 24752  df-pconn 34740
This theorem is referenced by:  qtoppconn  34755
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