| Step | Hyp | Ref
| Expression |
| 1 | | fundcmpsurinj.p |
. . 3
⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} |
| 2 | 1 | fundcmpsurbijinjpreimafv 47394 |
. 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) |
| 3 | | vex 3484 |
. . . . . . 7
⊢ 𝑗 ∈ V |
| 4 | | vex 3484 |
. . . . . . 7
⊢ 𝑓 ∈ V |
| 5 | 3, 4 | coex 7952 |
. . . . . 6
⊢ (𝑗 ∘ 𝑓) ∈ V |
| 6 | | simprl1 1219 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → 𝑔:𝐴–onto→𝑃) |
| 7 | | simp3 1139 |
. . . . . . . . 9
⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) |
| 8 | | f1of1 6847 |
. . . . . . . . . 10
⊢ (𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) → 𝑓:𝑃–1-1→(𝐹 “ 𝐴)) |
| 9 | 8 | 3ad2ant2 1135 |
. . . . . . . . 9
⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → 𝑓:𝑃–1-1→(𝐹 “ 𝐴)) |
| 10 | | f1co 6815 |
. . . . . . . . 9
⊢ ((𝑗:(𝐹 “ 𝐴)–1-1→𝐵 ∧ 𝑓:𝑃–1-1→(𝐹 “ 𝐴)) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵) |
| 11 | 7, 9, 10 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵) |
| 12 | 11 | ad2antrl 728 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → (𝑗 ∘ 𝑓):𝑃–1-1→𝐵) |
| 13 | | simprr 773 |
. . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) |
| 14 | 6, 12, 13 | 3jca 1129 |
. . . . . 6
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → (𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) |
| 15 | | f1eq1 6799 |
. . . . . . . 8
⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ:𝑃–1-1→𝐵 ↔ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵)) |
| 16 | | coeq1 5868 |
. . . . . . . . 9
⊢ (ℎ = (𝑗 ∘ 𝑓) → (ℎ ∘ 𝑔) = ((𝑗 ∘ 𝑓) ∘ 𝑔)) |
| 17 | 16 | eqeq2d 2748 |
. . . . . . . 8
⊢ (ℎ = (𝑗 ∘ 𝑓) → (𝐹 = (ℎ ∘ 𝑔) ↔ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) |
| 18 | 15, 17 | 3anbi23d 1441 |
. . . . . . 7
⊢ (ℎ = (𝑗 ∘ 𝑓) → ((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)) ↔ (𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)))) |
| 19 | 18 | spcegv 3597 |
. . . . . 6
⊢ ((𝑗 ∘ 𝑓) ∈ V → ((𝑔:𝐴–onto→𝑃 ∧ (𝑗 ∘ 𝑓):𝑃–1-1→𝐵 ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
| 20 | 5, 14, 19 | mpsyl 68 |
. . . . 5
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) ∧ ((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔))) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |
| 21 | 20 | ex 412 |
. . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
| 22 | 21 | exlimdvv 1934 |
. . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
| 23 | 22 | eximdv 1917 |
. 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (∃𝑔∃𝑓∃𝑗((𝑔:𝐴–onto→𝑃 ∧ 𝑓:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑗:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑗 ∘ 𝑓) ∘ 𝑔)) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))) |
| 24 | 2, 23 | mpd 15 |
1
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔))) |