| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | pwssplit1.y | . . 3
⊢ 𝑌 = (𝑊 ↑s 𝑈) | 
| 2 |  | pwssplit1.z | . . 3
⊢ 𝑍 = (𝑊 ↑s 𝑉) | 
| 3 |  | pwssplit1.b | . . 3
⊢ 𝐵 = (Base‘𝑌) | 
| 4 |  | pwssplit1.c | . . 3
⊢ 𝐶 = (Base‘𝑍) | 
| 5 |  | pwssplit1.f | . . 3
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) | 
| 6 | 1, 2, 3, 4, 5 | pwssplit0 21057 | . 2
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵⟶𝐶) | 
| 7 |  | simp1 1137 | . . . . . . . . 9
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑊 ∈ Mnd) | 
| 8 |  | simp2 1138 | . . . . . . . . . 10
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑈 ∈ 𝑋) | 
| 9 |  | simp3 1139 | . . . . . . . . . 10
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ⊆ 𝑈) | 
| 10 | 8, 9 | ssexd 5324 | . . . . . . . . 9
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝑉 ∈ V) | 
| 11 |  | eqid 2737 | . . . . . . . . . 10
⊢
(Base‘𝑊) =
(Base‘𝑊) | 
| 12 | 2, 11, 4 | pwselbasb 17533 | . . . . . . . . 9
⊢ ((𝑊 ∈ Mnd ∧ 𝑉 ∈ V) → (𝑎 ∈ 𝐶 ↔ 𝑎:𝑉⟶(Base‘𝑊))) | 
| 13 | 7, 10, 12 | syl2anc 584 | . . . . . . . 8
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → (𝑎 ∈ 𝐶 ↔ 𝑎:𝑉⟶(Base‘𝑊))) | 
| 14 | 13 | biimpa 476 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → 𝑎:𝑉⟶(Base‘𝑊)) | 
| 15 |  | fvex 6919 | . . . . . . . . . 10
⊢
(0g‘𝑊) ∈ V | 
| 16 | 15 | fconst 6794 | . . . . . . . . 9
⊢ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}):(𝑈 ∖ 𝑉)⟶{(0g‘𝑊)} | 
| 17 | 16 | a1i 11 | . . . . . . . 8
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}):(𝑈 ∖ 𝑉)⟶{(0g‘𝑊)}) | 
| 18 |  | simpl1 1192 | . . . . . . . . . 10
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → 𝑊 ∈ Mnd) | 
| 19 |  | eqid 2737 | . . . . . . . . . . 11
⊢
(0g‘𝑊) = (0g‘𝑊) | 
| 20 | 11, 19 | mndidcl 18762 | . . . . . . . . . 10
⊢ (𝑊 ∈ Mnd →
(0g‘𝑊)
∈ (Base‘𝑊)) | 
| 21 | 18, 20 | syl 17 | . . . . . . . . 9
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (0g‘𝑊) ∈ (Base‘𝑊)) | 
| 22 | 21 | snssd 4809 | . . . . . . . 8
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → {(0g‘𝑊)} ⊆ (Base‘𝑊)) | 
| 23 | 17, 22 | fssd 6753 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}):(𝑈 ∖ 𝑉)⟶(Base‘𝑊)) | 
| 24 |  | disjdif 4472 | . . . . . . . 8
⊢ (𝑉 ∩ (𝑈 ∖ 𝑉)) = ∅ | 
| 25 | 24 | a1i 11 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑉 ∩ (𝑈 ∖ 𝑉)) = ∅) | 
| 26 |  | fun 6770 | . . . . . . 7
⊢ (((𝑎:𝑉⟶(Base‘𝑊) ∧ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}):(𝑈 ∖ 𝑉)⟶(Base‘𝑊)) ∧ (𝑉 ∩ (𝑈 ∖ 𝑉)) = ∅) → (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):(𝑉 ∪ (𝑈 ∖ 𝑉))⟶((Base‘𝑊) ∪ (Base‘𝑊))) | 
| 27 | 14, 23, 25, 26 | syl21anc 838 | . . . . . 6
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):(𝑉 ∪ (𝑈 ∖ 𝑉))⟶((Base‘𝑊) ∪ (Base‘𝑊))) | 
| 28 |  | simpl3 1194 | . . . . . . . 8
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → 𝑉 ⊆ 𝑈) | 
| 29 |  | undif 4482 | . . . . . . . 8
⊢ (𝑉 ⊆ 𝑈 ↔ (𝑉 ∪ (𝑈 ∖ 𝑉)) = 𝑈) | 
| 30 | 28, 29 | sylib 218 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑉 ∪ (𝑈 ∖ 𝑉)) = 𝑈) | 
| 31 |  | unidm 4157 | . . . . . . . 8
⊢
((Base‘𝑊)
∪ (Base‘𝑊)) =
(Base‘𝑊) | 
| 32 | 31 | a1i 11 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((Base‘𝑊) ∪ (Base‘𝑊)) = (Base‘𝑊)) | 
| 33 | 30, 32 | feq23d 6731 | . . . . . 6
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):(𝑉 ∪ (𝑈 ∖ 𝑉))⟶((Base‘𝑊) ∪ (Base‘𝑊)) ↔ (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):𝑈⟶(Base‘𝑊))) | 
| 34 | 27, 33 | mpbid 232 | . . . . 5
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):𝑈⟶(Base‘𝑊)) | 
| 35 |  | simpl2 1193 | . . . . . 6
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → 𝑈 ∈ 𝑋) | 
| 36 | 1, 11, 3 | pwselbasb 17533 | . . . . . 6
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋) → ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ∈ 𝐵 ↔ (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):𝑈⟶(Base‘𝑊))) | 
| 37 | 18, 35, 36 | syl2anc 584 | . . . . 5
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ∈ 𝐵 ↔ (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})):𝑈⟶(Base‘𝑊))) | 
| 38 | 34, 37 | mpbird 257 | . . . 4
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ∈ 𝐵) | 
| 39 | 5 | fvtresfn 7018 | . . . . . 6
⊢ ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ∈ 𝐵 → (𝐹‘(𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}))) = ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ↾ 𝑉)) | 
| 40 | 38, 39 | syl 17 | . . . . 5
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝐹‘(𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}))) = ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ↾ 𝑉)) | 
| 41 |  | resundir 6012 | . . . . . . 7
⊢ ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ↾ 𝑉) = ((𝑎 ↾ 𝑉) ∪ (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) ↾ 𝑉)) | 
| 42 |  | ffn 6736 | . . . . . . . . 9
⊢ (𝑎:𝑉⟶(Base‘𝑊) → 𝑎 Fn 𝑉) | 
| 43 |  | fnresdm 6687 | . . . . . . . . 9
⊢ (𝑎 Fn 𝑉 → (𝑎 ↾ 𝑉) = 𝑎) | 
| 44 | 14, 42, 43 | 3syl 18 | . . . . . . . 8
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (𝑎 ↾ 𝑉) = 𝑎) | 
| 45 |  | disjdifr 4473 | . . . . . . . . 9
⊢ ((𝑈 ∖ 𝑉) ∩ 𝑉) = ∅ | 
| 46 |  | fnconstg 6796 | . . . . . . . . . . 11
⊢
((0g‘𝑊) ∈ V → ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) Fn (𝑈 ∖ 𝑉)) | 
| 47 | 15, 46 | ax-mp 5 | . . . . . . . . . 10
⊢ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) Fn (𝑈 ∖ 𝑉) | 
| 48 |  | fnresdisj 6688 | . . . . . . . . . 10
⊢ (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) Fn (𝑈 ∖ 𝑉) → (((𝑈 ∖ 𝑉) ∩ 𝑉) = ∅ ↔ (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) ↾ 𝑉) = ∅)) | 
| 49 | 47, 48 | mp1i 13 | . . . . . . . . 9
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (((𝑈 ∖ 𝑉) ∩ 𝑉) = ∅ ↔ (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) ↾ 𝑉) = ∅)) | 
| 50 | 45, 49 | mpbii 233 | . . . . . . . 8
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) ↾ 𝑉) = ∅) | 
| 51 | 44, 50 | uneq12d 4169 | . . . . . . 7
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑎 ↾ 𝑉) ∪ (((𝑈 ∖ 𝑉) × {(0g‘𝑊)}) ↾ 𝑉)) = (𝑎 ∪ ∅)) | 
| 52 | 41, 51 | eqtrid 2789 | . . . . . 6
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ↾ 𝑉) = (𝑎 ∪ ∅)) | 
| 53 |  | un0 4394 | . . . . . 6
⊢ (𝑎 ∪ ∅) = 𝑎 | 
| 54 | 52, 53 | eqtrdi 2793 | . . . . 5
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ↾ 𝑉) = 𝑎) | 
| 55 | 40, 54 | eqtr2d 2778 | . . . 4
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → 𝑎 = (𝐹‘(𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})))) | 
| 56 |  | fveq2 6906 | . . . . 5
⊢ (𝑏 = (𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) → (𝐹‘𝑏) = (𝐹‘(𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})))) | 
| 57 | 56 | rspceeqv 3645 | . . . 4
⊢ (((𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})) ∈ 𝐵 ∧ 𝑎 = (𝐹‘(𝑎 ∪ ((𝑈 ∖ 𝑉) × {(0g‘𝑊)})))) → ∃𝑏 ∈ 𝐵 𝑎 = (𝐹‘𝑏)) | 
| 58 | 38, 55, 57 | syl2anc 584 | . . 3
⊢ (((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) ∧ 𝑎 ∈ 𝐶) → ∃𝑏 ∈ 𝐵 𝑎 = (𝐹‘𝑏)) | 
| 59 | 58 | ralrimiva 3146 | . 2
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐵 𝑎 = (𝐹‘𝑏)) | 
| 60 |  | dffo3 7122 | . 2
⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑎 ∈ 𝐶 ∃𝑏 ∈ 𝐵 𝑎 = (𝐹‘𝑏))) | 
| 61 | 6, 59, 60 | sylanbrc 583 | 1
⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵–onto→𝐶) |