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Theorem cnpconn 34209
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 22736 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1135 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 eqid 2732 . . . . . . . . 9 𝐽 = 𝐽
43pconncn 34203 . . . . . . . 8 ((𝐽 ∈ PConn ∧ 𝑢 𝐽𝑣 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
543expb 1120 . . . . . . 7 ((𝐽 ∈ PConn ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
653ad2antl1 1185 . . . . . 6 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7 simprl 769 . . . . . . . 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 1214 . . . . . . . 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 22761 . . . . . . . 8 ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹𝑔) ∈ (II Cn 𝐾))
107, 8, 9syl2anc 584 . . . . . . 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 24388 . . . . . . . . . . 11 (0[,]1) = II
1211, 3cnf 22741 . . . . . . . . . 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 13442 . . . . . . . . 9 0 ∈ (0[,]1)
15 fvco3 6987 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹𝑔)‘0) = (𝐹‘(𝑔‘0)))
1613, 14, 15sylancl 586 . . . . . . . 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 779 . . . . . . . . 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 6892 . . . . . . . 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 2772 . . . . . . 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 13443 . . . . . . . . 9 1 ∈ (0[,]1)
21 fvco3 6987 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑔)‘1) = (𝐹‘(𝑔‘1)))
2213, 20, 21sylancl 586 . . . . . . . 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 780 . . . . . . . . 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 6892 . . . . . . . 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 2772 . . . . . . 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 6887 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘0) = ((𝐹𝑔)‘0))
2726eqeq1d 2734 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘0) = (𝐹𝑢) ↔ ((𝐹𝑔)‘0) = (𝐹𝑢)))
28 fveq1 6887 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘1) = ((𝐹𝑔)‘1))
2928eqeq1d 2734 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘1) = (𝐹𝑣) ↔ ((𝐹𝑔)‘1) = (𝐹𝑣)))
3027, 29anbi12d 631 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ (((𝐹𝑔)‘0) = (𝐹𝑢) ∧ ((𝐹𝑔)‘1) = (𝐹𝑣))))
3130rspcev 3612 . . . . . . 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 835 . . . . . 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 3161 . . . . 5 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
3433ralrimivva 3200 . . . 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 22741 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
37363ad2ant3 1135 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
38 forn 6805 . . . . . . . 8 (𝐹:𝑋onto𝑌 → ran 𝐹 = 𝑌)
39383ad2ant2 1134 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40 dffo2 6806 . . . . . . 7 (𝐹: 𝐽onto𝑌 ↔ (𝐹: 𝐽𝑌 ∧ ran 𝐹 = 𝑌))
4137, 39, 40sylanbrc 583 . . . . . 6 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto𝑌)
42 eqeq2 2744 . . . . . . . . 9 ((𝐹𝑣) = 𝑦 → ((𝑓‘1) = (𝐹𝑣) ↔ (𝑓‘1) = 𝑦))
4342anbi2d 629 . . . . . . . 8 ((𝐹𝑣) = 𝑦 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4443rexbidv 3178 . . . . . . 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 3177 . . . 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 2744 . . . . . . . 8 ((𝐹𝑢) = 𝑥 → ((𝑓‘0) = (𝐹𝑢) ↔ (𝑓‘0) = 𝑥))
5049anbi1d 630 . . . . . . 7 ((𝐹𝑢) = 𝑥 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5150rexbidv 3178 . . . . . 6 ((𝐹𝑢) = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5251ralbidv 3177 . . . . 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 34202 . 2 (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
572, 55, 56sylanbrc 583 1 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070   cuni 4907  ran crn 5676  ccom 5679  wf 6536  ontowfo 6538  cfv 6540  (class class class)co 7405  0cc0 11106  1c1 11107  [,]cicc 13323  Topctop 22386   Cn ccn 22719  IIcii 24382  PConncpconn 34198
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-icc 13327  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-ii 24384  df-pconn 34200
This theorem is referenced by:  qtoppconn  34215
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