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Theorem cnpconn 31592
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 21325 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1165 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 eqid 2765 . . . . . . . . 9 𝐽 = 𝐽
43pconncn 31586 . . . . . . . 8 ((𝐽 ∈ PConn ∧ 𝑢 𝐽𝑣 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
543expb 1149 . . . . . . 7 ((𝐽 ∈ PConn ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
653ad2antl1 1236 . . . . . 6 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7 simprl 787 . . . . . . . 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 1273 . . . . . . . 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 21350 . . . . . . . 8 ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹𝑔) ∈ (II Cn 𝐾))
107, 8, 9syl2anc 579 . . . . . . 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 22963 . . . . . . . . . . 11 (0[,]1) = II
1211, 3cnf 21330 . . . . . . . . . 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 12495 . . . . . . . . 9 0 ∈ (0[,]1)
15 fvco3 6464 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹𝑔)‘0) = (𝐹‘(𝑔‘0)))
1613, 14, 15sylancl 580 . . . . . . . 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 799 . . . . . . . . 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 6379 . . . . . . . 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 2799 . . . . . . 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 12496 . . . . . . . . 9 1 ∈ (0[,]1)
21 fvco3 6464 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑔)‘1) = (𝐹‘(𝑔‘1)))
2213, 20, 21sylancl 580 . . . . . . . 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 800 . . . . . . . . 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 6379 . . . . . . . 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 2799 . . . . . . 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 6374 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘0) = ((𝐹𝑔)‘0))
2726eqeq1d 2767 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘0) = (𝐹𝑢) ↔ ((𝐹𝑔)‘0) = (𝐹𝑢)))
28 fveq1 6374 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘1) = ((𝐹𝑔)‘1))
2928eqeq1d 2767 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘1) = (𝐹𝑣) ↔ ((𝐹𝑔)‘1) = (𝐹𝑣)))
3027, 29anbi12d 624 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ (((𝐹𝑔)‘0) = (𝐹𝑢) ∧ ((𝐹𝑔)‘1) = (𝐹𝑣))))
3130rspcev 3461 . . . . . . 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 865 . . . . . 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 3182 . . . . 5 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
3433ralrimivva 3118 . . . 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 21330 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
37363ad2ant3 1165 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
38 forn 6301 . . . . . . . 8 (𝐹:𝑋onto𝑌 → ran 𝐹 = 𝑌)
39383ad2ant2 1164 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40 dffo2 6302 . . . . . . 7 (𝐹: 𝐽onto𝑌 ↔ (𝐹: 𝐽𝑌 ∧ ran 𝐹 = 𝑌))
4137, 39, 40sylanbrc 578 . . . . . 6 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto𝑌)
42 eqeq2 2776 . . . . . . . . 9 ((𝐹𝑣) = 𝑦 → ((𝑓‘1) = (𝐹𝑣) ↔ (𝑓‘1) = 𝑦))
4342anbi2d 622 . . . . . . . 8 ((𝐹𝑣) = 𝑦 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4443rexbidv 3199 . . . . . . 7 ((𝐹𝑣) = 𝑦 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4544cbvfo 6736 . . . . . 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 3133 . . . 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 223 . . 3 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦))
49 eqeq2 2776 . . . . . . . 8 ((𝐹𝑢) = 𝑥 → ((𝑓‘0) = (𝐹𝑢) ↔ (𝑓‘0) = 𝑥))
5049anbi1d 623 . . . . . . 7 ((𝐹𝑢) = 𝑥 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5150rexbidv 3199 . . . . . 6 ((𝐹𝑢) = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5251ralbidv 3133 . . . . 5 ((𝐹𝑢) = 𝑥 → (∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5352cbvfo 6736 . . . 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 223 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
5635ispconn 31585 . 2 (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
572, 55, 56sylanbrc 578 1 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056   cuni 4594  ran crn 5278  ccom 5281  wf 6064  ontowfo 6066  cfv 6068  (class class class)co 6842  0cc0 10189  1c1 10190  [,]cicc 12380  Topctop 20977   Cn ccn 21308  IIcii 22957  PConncpconn 31581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-icc 12384  df-seq 13009  df-exp 13068  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-topgen 16372  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-top 20978  df-topon 20995  df-bases 21030  df-cn 21311  df-ii 22959  df-pconn 31583
This theorem is referenced by:  qtoppconn  31598
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