| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sconntop 35234 | . . . . . . . . 9
⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) | 
| 2 | 1 | adantl 481 | . . . . . . . 8
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → 𝐽 ∈ Top) | 
| 3 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → 𝑌 ∈ 𝑋) | 
| 4 |  | eqid 2736 | . . . . . . . . 9
⊢ (𝐽 π1 𝑌) = (𝐽 π1 𝑌) | 
| 5 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘(𝐽
π1 𝑌)) =
(Base‘(𝐽
π1 𝑌)) | 
| 6 |  | simpl 482 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑋) → 𝐽 ∈ Top) | 
| 7 |  | sconnpi1.1 | . . . . . . . . . . 11
⊢ 𝑋 = ∪
𝐽 | 
| 8 | 7 | toptopon 22924 | . . . . . . . . . 10
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | 
| 9 | 6, 8 | sylib 218 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 10 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋) | 
| 11 | 4, 5, 9, 10 | elpi1 25079 | . . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝑋) → (𝑥 ∈ (Base‘(𝐽 π1 𝑌)) ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝑥 = [𝑓]( ≃ph‘𝐽)))) | 
| 12 | 2, 3, 11 | syl2anc 584 | . . . . . . 7
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (𝑥 ∈ (Base‘(𝐽 π1 𝑌)) ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝑥 = [𝑓]( ≃ph‘𝐽)))) | 
| 13 |  | phtpcer 25028 | . . . . . . . . . . . . 13
⊢ (
≃ph‘𝐽) Er (II Cn 𝐽) | 
| 14 | 13 | a1i 11 | . . . . . . . . . . . 12
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → (
≃ph‘𝐽) Er (II Cn 𝐽)) | 
| 15 |  | simpllr 775 | . . . . . . . . . . . . . 14
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → 𝐽 ∈ SConn) | 
| 16 |  | simplr 768 | . . . . . . . . . . . . . 14
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → 𝑓 ∈ (II Cn 𝐽)) | 
| 17 |  | simprl 770 | . . . . . . . . . . . . . . 15
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → (𝑓‘0) = 𝑌) | 
| 18 |  | simprr 772 | . . . . . . . . . . . . . . 15
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → (𝑓‘1) = 𝑌) | 
| 19 | 17, 18 | eqtr4d 2779 | . . . . . . . . . . . . . 14
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → (𝑓‘0) = (𝑓‘1)) | 
| 20 |  | sconnpht 35235 | . . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ SConn ∧ 𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1)) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) | 
| 21 | 15, 16, 19, 20 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) | 
| 22 | 17 | sneqd 4637 | . . . . . . . . . . . . . 14
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → {(𝑓‘0)} = {𝑌}) | 
| 23 | 22 | xpeq2d 5714 | . . . . . . . . . . . . 13
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) ×
{𝑌})) | 
| 24 | 21, 23 | breqtrd 5168 | . . . . . . . . . . . 12
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → 𝑓( ≃ph‘𝐽)((0[,]1) × {𝑌})) | 
| 25 | 14, 24 | erthi 8799 | . . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → [𝑓]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})](
≃ph‘𝐽)) | 
| 26 | 2, 8 | sylib 218 | . . . . . . . . . . . . 13
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 27 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ ((0[,]1)
× {𝑌}) = ((0[,]1)
× {𝑌}) | 
| 28 | 4, 27 | pi1id 25085 | . . . . . . . . . . . . 13
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝑋) → [((0[,]1) × {𝑌})](
≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌))) | 
| 29 | 26, 3, 28 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → [((0[,]1) ×
{𝑌})](
≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌))) | 
| 30 | 29 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → [((0[,]1) × {𝑌})](
≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌))) | 
| 31 | 25, 30 | eqtrd 2776 | . . . . . . . . . 10
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → [𝑓]( ≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌))) | 
| 32 |  | velsn 4641 | . . . . . . . . . . 11
⊢ (𝑥 ∈
{(0g‘(𝐽
π1 𝑌))}
↔ 𝑥 =
(0g‘(𝐽
π1 𝑌))) | 
| 33 |  | eqeq1 2740 | . . . . . . . . . . 11
⊢ (𝑥 = [𝑓]( ≃ph‘𝐽) → (𝑥 = (0g‘(𝐽 π1 𝑌)) ↔ [𝑓]( ≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌)))) | 
| 34 | 32, 33 | bitrid 283 | . . . . . . . . . 10
⊢ (𝑥 = [𝑓]( ≃ph‘𝐽) → (𝑥 ∈ {(0g‘(𝐽 π1 𝑌))} ↔ [𝑓]( ≃ph‘𝐽) = (0g‘(𝐽 π1 𝑌)))) | 
| 35 | 31, 34 | syl5ibrcom 247 | . . . . . . . . 9
⊢ ((((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) ∧ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)) → (𝑥 = [𝑓]( ≃ph‘𝐽) → 𝑥 ∈ {(0g‘(𝐽 π1 𝑌))})) | 
| 36 | 35 | expimpd 453 | . . . . . . . 8
⊢ (((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) ∧ 𝑓 ∈ (II Cn 𝐽)) → ((((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝑥 = [𝑓]( ≃ph‘𝐽)) → 𝑥 ∈ {(0g‘(𝐽 π1 𝑌))})) | 
| 37 | 36 | rexlimdva 3154 | . . . . . . 7
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝑥 = [𝑓]( ≃ph‘𝐽)) → 𝑥 ∈ {(0g‘(𝐽 π1 𝑌))})) | 
| 38 | 12, 37 | sylbid 240 | . . . . . 6
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (𝑥 ∈ (Base‘(𝐽 π1 𝑌)) → 𝑥 ∈ {(0g‘(𝐽 π1 𝑌))})) | 
| 39 | 38 | ssrdv 3988 | . . . . 5
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (Base‘(𝐽 π1 𝑌)) ⊆
{(0g‘(𝐽
π1 𝑌))}) | 
| 40 | 4 | pi1grp 25084 | . . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝑋) → (𝐽 π1 𝑌) ∈ Grp) | 
| 41 | 26, 3, 40 | syl2anc 584 | . . . . . . 7
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (𝐽 π1 𝑌) ∈ Grp) | 
| 42 |  | eqid 2736 | . . . . . . . 8
⊢
(0g‘(𝐽 π1 𝑌)) = (0g‘(𝐽 π1 𝑌)) | 
| 43 | 5, 42 | grpidcl 18984 | . . . . . . 7
⊢ ((𝐽 π1 𝑌) ∈ Grp →
(0g‘(𝐽
π1 𝑌)) ∈
(Base‘(𝐽
π1 𝑌))) | 
| 44 | 41, 43 | syl 17 | . . . . . 6
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) →
(0g‘(𝐽
π1 𝑌)) ∈
(Base‘(𝐽
π1 𝑌))) | 
| 45 | 44 | snssd 4808 | . . . . 5
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) →
{(0g‘(𝐽
π1 𝑌))}
⊆ (Base‘(𝐽
π1 𝑌))) | 
| 46 | 39, 45 | eqssd 4000 | . . . 4
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (Base‘(𝐽 π1 𝑌)) =
{(0g‘(𝐽
π1 𝑌))}) | 
| 47 |  | fvex 6918 | . . . . 5
⊢
(0g‘(𝐽 π1 𝑌)) ∈ V | 
| 48 | 47 | ensn1 9062 | . . . 4
⊢
{(0g‘(𝐽 π1 𝑌))} ≈ 1o | 
| 49 | 46, 48 | eqbrtrdi 5181 | . . 3
⊢ ((𝑌 ∈ 𝑋 ∧ 𝐽 ∈ SConn) → (Base‘(𝐽 π1 𝑌)) ≈
1o) | 
| 50 | 49 | adantll 714 | . 2
⊢ (((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ 𝐽 ∈ SConn) → (Base‘(𝐽 π1 𝑌)) ≈
1o) | 
| 51 |  | simpll 766 | . . 3
⊢ (((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) → 𝐽 ∈ PConn) | 
| 52 |  | eqid 2736 | . . . . . . . . 9
⊢ (𝐽 π1 (𝑓‘0)) = (𝐽 π1 (𝑓‘0)) | 
| 53 |  | eqid 2736 | . . . . . . . . 9
⊢
(Base‘(𝐽
π1 (𝑓‘0))) = (Base‘(𝐽 π1 (𝑓‘0))) | 
| 54 |  | simplll 774 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ PConn) | 
| 55 |  | pconntop 35231 | . . . . . . . . . . 11
⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | 
| 56 | 54, 55 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ Top) | 
| 57 | 56, 8 | sylib 218 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 58 |  | simprl 770 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓 ∈ (II Cn 𝐽)) | 
| 59 |  | iiuni 24908 | . . . . . . . . . . . 12
⊢ (0[,]1) =
∪ II | 
| 60 | 59, 7 | cnf 23255 | . . . . . . . . . . 11
⊢ (𝑓 ∈ (II Cn 𝐽) → 𝑓:(0[,]1)⟶𝑋) | 
| 61 | 58, 60 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓:(0[,]1)⟶𝑋) | 
| 62 |  | 0elunit 13510 | . . . . . . . . . 10
⊢ 0 ∈
(0[,]1) | 
| 63 |  | ffvelcdm 7100 | . . . . . . . . . 10
⊢ ((𝑓:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) →
(𝑓‘0) ∈ 𝑋) | 
| 64 | 61, 62, 63 | sylancl 586 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) ∈ 𝑋) | 
| 65 |  | eqidd 2737 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘0)) | 
| 66 |  | simprr 772 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘0) = (𝑓‘1)) | 
| 67 | 66 | eqcomd 2742 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓‘1) = (𝑓‘0)) | 
| 68 | 52, 53, 57, 64, 58, 65, 67 | elpi1i 25080 | . . . . . . . 8
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → [𝑓]( ≃ph‘𝐽) ∈ (Base‘(𝐽 π1 (𝑓‘0)))) | 
| 69 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢ ((0[,]1)
× {(𝑓‘0)}) =
((0[,]1) × {(𝑓‘0)}) | 
| 70 | 69 | pcoptcl 25055 | . . . . . . . . . . . 12
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑓‘0) ∈ 𝑋) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐽) ∧ (((0[,]1) ×
{(𝑓‘0)})‘0) =
(𝑓‘0) ∧ (((0[,]1)
× {(𝑓‘0)})‘1) = (𝑓‘0))) | 
| 71 | 57, 64, 70 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐽) ∧ (((0[,]1) ×
{(𝑓‘0)})‘0) =
(𝑓‘0) ∧ (((0[,]1)
× {(𝑓‘0)})‘1) = (𝑓‘0))) | 
| 72 | 71 | simp1d 1142 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → ((0[,]1) × {(𝑓‘0)}) ∈ (II Cn 𝐽)) | 
| 73 | 71 | simp2d 1143 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (((0[,]1) × {(𝑓‘0)})‘0) = (𝑓‘0)) | 
| 74 | 71 | simp3d 1144 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (((0[,]1) × {(𝑓‘0)})‘1) = (𝑓‘0)) | 
| 75 | 52, 53, 57, 64, 72, 73, 74 | elpi1i 25080 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → [((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽) ∈ (Base‘(𝐽 π1 (𝑓‘0)))) | 
| 76 |  | simpllr 775 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑌 ∈ 𝑋) | 
| 77 | 7, 52, 4, 53, 5 | pconnpi1 35243 | . . . . . . . . . . . 12
⊢ ((𝐽 ∈ PConn ∧ (𝑓‘0) ∈ 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐽 π1 (𝑓‘0)) ≃𝑔 (𝐽 π1 𝑌)) | 
| 78 | 54, 64, 76, 77 | syl3anc 1372 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝐽 π1 (𝑓‘0)) ≃𝑔 (𝐽 π1 𝑌)) | 
| 79 | 53, 5 | gicen 19297 | . . . . . . . . . . 11
⊢ ((𝐽 π1 (𝑓‘0))
≃𝑔 (𝐽 π1 𝑌) → (Base‘(𝐽 π1 (𝑓‘0))) ≈ (Base‘(𝐽 π1 𝑌))) | 
| 80 | 78, 79 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (Base‘(𝐽 π1 (𝑓‘0))) ≈
(Base‘(𝐽
π1 𝑌))) | 
| 81 |  | simplr 768 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (Base‘(𝐽 π1 𝑌)) ≈
1o) | 
| 82 |  | entr 9047 | . . . . . . . . . 10
⊢
(((Base‘(𝐽
π1 (𝑓‘0))) ≈ (Base‘(𝐽 π1 𝑌)) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) →
(Base‘(𝐽
π1 (𝑓‘0))) ≈
1o) | 
| 83 | 80, 81, 82 | syl2anc 584 | . . . . . . . . 9
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (Base‘(𝐽 π1 (𝑓‘0))) ≈
1o) | 
| 84 |  | en1eqsn 9309 | . . . . . . . . 9
⊢
(([((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽) ∈ (Base‘(𝐽 π1 (𝑓‘0))) ∧ (Base‘(𝐽 π1 (𝑓‘0))) ≈
1o) → (Base‘(𝐽 π1 (𝑓‘0))) = {[((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽)}) | 
| 85 | 75, 83, 84 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (Base‘(𝐽 π1 (𝑓‘0))) = {[((0[,]1) ×
{(𝑓‘0)})](
≃ph‘𝐽)}) | 
| 86 | 68, 85 | eleqtrd 2842 | . . . . . . 7
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → [𝑓]( ≃ph‘𝐽) ∈ {[((0[,]1) ×
{(𝑓‘0)})](
≃ph‘𝐽)}) | 
| 87 |  | elsni 4642 | . . . . . . 7
⊢ ([𝑓](
≃ph‘𝐽) ∈ {[((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽)} → [𝑓]( ≃ph‘𝐽) = [((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽)) | 
| 88 | 86, 87 | syl 17 | . . . . . 6
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → [𝑓]( ≃ph‘𝐽) = [((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽)) | 
| 89 | 13 | a1i 11 | . . . . . . 7
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (
≃ph‘𝐽) Er (II Cn 𝐽)) | 
| 90 | 89, 58 | erth 8797 | . . . . . 6
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ [𝑓](
≃ph‘𝐽) = [((0[,]1) × {(𝑓‘0)})](
≃ph‘𝐽))) | 
| 91 | 88, 90 | mpbird 257 | . . . . 5
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝑓‘1))) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) | 
| 92 | 91 | expr 456 | . . . 4
⊢ ((((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) ∧ 𝑓 ∈ (II Cn 𝐽)) → ((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))) | 
| 93 | 92 | ralrimiva 3145 | . . 3
⊢ (((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) →
∀𝑓 ∈ (II Cn
𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}))) | 
| 94 |  | issconn 35232 | . . 3
⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧
∀𝑓 ∈ (II Cn
𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | 
| 95 | 51, 93, 94 | sylanbrc 583 | . 2
⊢ (((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) ∧ (Base‘(𝐽 π1 𝑌)) ≈ 1o) → 𝐽 ∈ SConn) | 
| 96 | 50, 95 | impbida 800 | 1
⊢ ((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈
1o)) |