| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) |
| 2 | 1 | mptexd 7244 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V) |
| 3 | | fundcmpsurinj.p |
. . . . . 6
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
| 4 | 3 | setpreimafvex 47370 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V) |
| 5 | 4 | adantl 481 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝑃 ∈ V) |
| 6 | 5 | mptexd 7244 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V) |
| 7 | | ffun 6739 |
. . . . 5
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) |
| 8 | | funimaexg 6653 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V) |
| 9 | 7, 8 | sylan 580 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V) |
| 10 | 9 | resiexd 7236 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)) ∈ V) |
| 11 | 2, 6, 10 | 3jca 1129 |
. 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V)) |
| 12 | | ffn 6736 |
. . . . 5
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) |
| 13 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) |
| 14 | 13 | sneqd 4638 |
. . . . . . . 8
⊢ (𝑎 = 𝑥 → {(𝐹‘𝑎)} = {(𝐹‘𝑥)}) |
| 15 | 14 | imaeq2d 6078 |
. . . . . . 7
⊢ (𝑎 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑎)}) = (◡𝐹 “ {(𝐹‘𝑥)})) |
| 16 | 15 | cbvmptv 5255 |
. . . . . 6
⊢ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)})) |
| 17 | 3, 16 | fundcmpsurinjlem2 47386 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃) |
| 18 | 12, 17 | sylan 580 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃) |
| 19 | | eqid 2737 |
. . . . . 6
⊢ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) |
| 20 | 3, 19 | imasetpreimafvbij 47393 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴)) |
| 21 | 12, 20 | sylan 580 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴)) |
| 22 | | f1oi 6886 |
. . . . . 6
⊢ ( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) |
| 23 | | f1of1 6847 |
. . . . . 6
⊢ (( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴)) |
| 24 | | fimass 6756 |
. . . . . . . 8
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵) |
| 25 | | f1ss 6809 |
. . . . . . . 8
⊢ ((( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) |
| 26 | 24, 25 | sylan2 593 |
. . . . . . 7
⊢ ((( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) ∧ 𝐹:𝐴⟶𝐵) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) |
| 27 | 26 | ex 412 |
. . . . . 6
⊢ (( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→(𝐹 “ 𝐴) → (𝐹:𝐴⟶𝐵 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)) |
| 28 | 22, 23, 27 | mp2b 10 |
. . . . 5
⊢ (𝐹:𝐴⟶𝐵 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) |
| 29 | 28 | adantr 480 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) |
| 30 | 18, 21, 29 | 3jca 1129 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)) |
| 31 | 12 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴) |
| 32 | | uniimaprimaeqfv 47369 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎)) |
| 33 | 31, 32 | sylan 580 |
. . . . . . . 8
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝐹‘𝑎)) |
| 34 | 33 | fveq2d 6910 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))) = (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))) |
| 35 | 34 | mpteq2dva 5242 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)))) |
| 36 | | ffrn 6749 |
. . . . . . . . . 10
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) |
| 37 | 36 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹:𝐴⟶ran 𝐹) |
| 38 | 37 | funfvima2d 7252 |
. . . . . . . 8
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ (𝐹 “ 𝐴)) |
| 39 | | fvresi 7193 |
. . . . . . . 8
⊢ ((𝐹‘𝑎) ∈ (𝐹 “ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)) = (𝐹‘𝑎)) |
| 40 | 38, 39 | syl 17 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎)) = (𝐹‘𝑎)) |
| 41 | 40 | mpteq2dva 5242 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘(𝐹‘𝑎))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎))) |
| 42 | 35, 41 | eqtrd 2777 |
. . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎))) |
| 43 | 12 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐹 Fn 𝐴) |
| 44 | 1 | adantr 480 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑉) |
| 45 | | simpr 484 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) |
| 46 | 3 | preimafvelsetpreimafv 47375 |
. . . . . . 7
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃) |
| 47 | 43, 44, 45, 46 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ 𝑎 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑎)}) ∈ 𝑃) |
| 48 | | eqidd 2738 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) |
| 49 | | eqidd 2738 |
. . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦)))) |
| 50 | | imaeq2 6074 |
. . . . . . . 8
⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))) |
| 51 | 50 | unieqd 4920 |
. . . . . . 7
⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → ∪
(𝐹 “ 𝑦) = ∪
(𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))) |
| 52 | 51 | fveq2d 6910 |
. . . . . 6
⊢ (𝑦 = (◡𝐹 “ {(𝐹‘𝑎)}) → (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦)) = (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)})))) |
| 53 | 47, 48, 49, 52 | fmptco 7149 |
. . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) = (𝑎 ∈ 𝐴 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑎)}))))) |
| 54 | | dffn5 6967 |
. . . . . . 7
⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎))) |
| 55 | 12, 54 | sylib 218 |
. . . . . 6
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎))) |
| 56 | 55 | adantr 480 |
. . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎))) |
| 57 | 42, 53, 56 | 3eqtr4rd 2788 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) |
| 58 | | f1of 6848 |
. . . . . . . . . 10
⊢ (( I
↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1-onto→(𝐹 “ 𝐴) → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴)) |
| 59 | 22, 58 | mp1i 13 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝐴 → ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴)) |
| 60 | | fnima 6698 |
. . . . . . . . . . 11
⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹) |
| 61 | 60 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴)) |
| 62 | 61 | feq2d 6722 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝐴 → (( I ↾ (𝐹 “ 𝐴)):ran 𝐹⟶(𝐹 “ 𝐴) ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)⟶(𝐹 “ 𝐴))) |
| 63 | 59, 62 | mpbird 257 |
. . . . . . . 8
⊢ (𝐹 Fn 𝐴 → ( I ↾ (𝐹 “ 𝐴)):ran 𝐹⟶(𝐹 “ 𝐴)) |
| 64 | 3 | uniimaelsetpreimafv 47383 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝑃) → ∪ (𝐹 “ 𝑦) ∈ ran 𝐹) |
| 65 | 63, 64 | cofmpt 7152 |
. . . . . . 7
⊢ (𝐹 Fn 𝐴 → (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) = (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦)))) |
| 66 | 65 | eqcomd 2743 |
. . . . . 6
⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)))) |
| 67 | 31, 66 | syl 17 |
. . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)))) |
| 68 | 67 | coeq1d 5872 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝑃 ↦ (( I ↾ (𝐹 “ 𝐴))‘∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) |
| 69 | 57, 68 | eqtrd 2777 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) |
| 70 | 30, 69 | jca 511 |
. 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))) |
| 71 | | foeq1 6816 |
. . . . . 6
⊢ (𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) → (𝑔:𝐴–onto→𝑃 ↔ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)) |
| 72 | 71 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑔:𝐴–onto→𝑃 ↔ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃)) |
| 73 | | f1oeq1 6836 |
. . . . . 6
⊢ (ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) → (ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ↔ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))) |
| 74 | 73 | 3ad2ant2 1135 |
. . . . 5
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ↔ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴))) |
| 75 | | f1eq1 6799 |
. . . . . 6
⊢ (𝑖 = ( I ↾ (𝐹 “ 𝐴)) → (𝑖:(𝐹 “ 𝐴)–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)) |
| 76 | 75 | 3ad2ant3 1136 |
. . . . 5
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑖:(𝐹 “ 𝐴)–1-1→𝐵 ↔ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵)) |
| 77 | 72, 74, 76 | 3anbi123d 1438 |
. . . 4
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ↔ ((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵))) |
| 78 | | simp3 1139 |
. . . . . . 7
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → 𝑖 = ( I ↾ (𝐹 “ 𝐴))) |
| 79 | | simp2 1138 |
. . . . . . 7
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) |
| 80 | 78, 79 | coeq12d 5875 |
. . . . . 6
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝑖 ∘ ℎ) = (( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)))) |
| 81 | | simp1 1137 |
. . . . . 6
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → 𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))) |
| 82 | 80, 81 | coeq12d 5875 |
. . . . 5
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → ((𝑖 ∘ ℎ) ∘ 𝑔) = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) |
| 83 | 82 | eqeq2d 2748 |
. . . 4
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔) ↔ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)}))))) |
| 84 | 77, 83 | anbi12d 632 |
. . 3
⊢ ((𝑔 = (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∧ ℎ = (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∧ 𝑖 = ( I ↾ (𝐹 “ 𝐴))) → (((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))))) |
| 85 | 84 | spc3egv 3603 |
. 2
⊢ (((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})) ∈ V ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)) ∈ V ∧ ( I ↾ (𝐹 “ 𝐴)) ∈ V) → ((((𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})):𝐴–onto→𝑃 ∧ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦)):𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ ( I ↾ (𝐹 “ 𝐴)):(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((( I ↾ (𝐹 “ 𝐴)) ∘ (𝑦 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑦))) ∘ (𝑎 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑎)})))) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))) |
| 86 | 11, 70, 85 | sylc 65 |
1
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) |