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Theorem cnpconn 32698
 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 21931 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1133 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 eqid 2759 . . . . . . . . 9 𝐽 = 𝐽
43pconncn 32692 . . . . . . . 8 ((𝐽 ∈ PConn ∧ 𝑢 𝐽𝑣 𝐽) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
543expb 1118 . . . . . . 7 ((𝐽 ∈ PConn ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
653ad2antl1 1183 . . . . . 6 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑔 ∈ (II Cn 𝐽)((𝑔‘0) = 𝑢 ∧ (𝑔‘1) = 𝑣))
7 simprl 771 . . . . . . . 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 1212 . . . . . . . 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 21956 . . . . . . . 8 ((𝑔 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹𝑔) ∈ (II Cn 𝐾))
107, 8, 9syl2anc 588 . . . . . . 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 23572 . . . . . . . . . . 11 (0[,]1) = II
1211, 3cnf 21936 . . . . . . . . . 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 12891 . . . . . . . . 9 0 ∈ (0[,]1)
15 fvco3 6749 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 0 ∈ (0[,]1)) → ((𝐹𝑔)‘0) = (𝐹‘(𝑔‘0)))
1613, 14, 15sylancl 590 . . . . . . . 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 781 . . . . . . . . 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 6660 . . . . . . . 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 2794 . . . . . . 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 12892 . . . . . . . . 9 1 ∈ (0[,]1)
21 fvco3 6749 . . . . . . . . 9 ((𝑔:(0[,]1)⟶ 𝐽 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑔)‘1) = (𝐹‘(𝑔‘1)))
2213, 20, 21sylancl 590 . . . . . . . 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 782 . . . . . . . . 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 6660 . . . . . . . 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 2794 . . . . . . 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 6655 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘0) = ((𝐹𝑔)‘0))
2726eqeq1d 2761 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘0) = (𝐹𝑢) ↔ ((𝐹𝑔)‘0) = (𝐹𝑢)))
28 fveq1 6655 . . . . . . . . . 10 (𝑓 = (𝐹𝑔) → (𝑓‘1) = ((𝐹𝑔)‘1))
2928eqeq1d 2761 . . . . . . . . 9 (𝑓 = (𝐹𝑔) → ((𝑓‘1) = (𝐹𝑣) ↔ ((𝐹𝑔)‘1) = (𝐹𝑣)))
3027, 29anbi12d 634 . . . . . . . 8 (𝑓 = (𝐹𝑔) → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ (((𝐹𝑔)‘0) = (𝐹𝑢) ∧ ((𝐹𝑔)‘1) = (𝐹𝑣))))
3130rspcev 3542 . . . . . . 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 836 . . . . . 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 3216 . . . . 5 (((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢 𝐽𝑣 𝐽)) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)))
3433ralrimivva 3121 . . . 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 21936 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
37363ad2ant3 1133 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽𝑌)
38 forn 6577 . . . . . . . 8 (𝐹:𝑋onto𝑌 → ran 𝐹 = 𝑌)
39383ad2ant2 1132 . . . . . . 7 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ran 𝐹 = 𝑌)
40 dffo2 6578 . . . . . . 7 (𝐹: 𝐽onto𝑌 ↔ (𝐹: 𝐽𝑌 ∧ ran 𝐹 = 𝑌))
4137, 39, 40sylanbrc 587 . . . . . 6 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽onto𝑌)
42 eqeq2 2771 . . . . . . . . 9 ((𝐹𝑣) = 𝑦 → ((𝑓‘1) = (𝐹𝑣) ↔ (𝑓‘1) = 𝑦))
4342anbi2d 632 . . . . . . . 8 ((𝐹𝑣) = 𝑦 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4443rexbidv 3222 . . . . . . 7 ((𝐹𝑣) = 𝑦 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = (𝐹𝑣)) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦)))
4544cbvfo 7035 . . . . . 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 3127 . . . 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 235 . . 3 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 𝐽𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦))
49 eqeq2 2771 . . . . . . . 8 ((𝐹𝑢) = 𝑥 → ((𝑓‘0) = (𝐹𝑢) ↔ (𝑓‘0) = 𝑥))
5049anbi1d 633 . . . . . . 7 ((𝐹𝑢) = 𝑥 → (((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5150rexbidv 3222 . . . . . 6 ((𝐹𝑢) = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5251ralbidv 3127 . . . . 5 ((𝐹𝑢) = 𝑥 → (∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = (𝐹𝑢) ∧ (𝑓‘1) = 𝑦) ↔ ∀𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
5352cbvfo 7035 . . . 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 235 . 2 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
5635ispconn 32691 . 2 (𝐾 ∈ PConn ↔ (𝐾 ∈ Top ∧ ∀𝑥𝑌𝑦𝑌𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
572, 55, 56sylanbrc 587 1 ((𝐽 ∈ PConn ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  ∪ cuni 4796  ran crn 5523   ∘ ccom 5526  ⟶wf 6329  –onto→wfo 6331  ‘cfv 6333  (class class class)co 7148  0cc0 10565  1c1 10566  [,]cicc 12772  Topctop 21583   Cn ccn 21914  IIcii 23566  PConncpconn 32687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642  ax-pre-sup 10643 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-er 8297  df-map 8416  df-en 8526  df-dom 8527  df-sdom 8528  df-sup 8929  df-inf 8930  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-div 11326  df-nn 11665  df-2 11727  df-3 11728  df-n0 11925  df-z 12011  df-uz 12273  df-q 12379  df-rp 12421  df-xneg 12538  df-xadd 12539  df-xmul 12540  df-icc 12776  df-seq 13409  df-exp 13470  df-cj 14496  df-re 14497  df-im 14498  df-sqrt 14632  df-abs 14633  df-topgen 16765  df-psmet 20148  df-xmet 20149  df-met 20150  df-bl 20151  df-mopn 20152  df-top 21584  df-topon 21601  df-bases 21636  df-cn 21917  df-ii 23568  df-pconn 32689 This theorem is referenced by:  qtoppconn  32704
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