| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ffun 6739 | . . . 4
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | 
| 2 |  | funimaexg 6653 | . . . 4
⊢ ((Fun
𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V) | 
| 3 | 1, 2 | sylan 580 | . . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ∈ V) | 
| 4 |  | abrexexg 7985 | . . . 4
⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) | 
| 5 | 4 | adantl 481 | . . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) | 
| 6 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | 
| 7 | 6 | sneqd 4638 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → {(𝐹‘𝑦)} = {(𝐹‘𝑥)}) | 
| 8 | 7 | imaeq2d 6078 | . . . . . . 7
⊢ (𝑦 = 𝑥 → (◡𝐹 “ {(𝐹‘𝑦)}) = (◡𝐹 “ {(𝐹‘𝑥)})) | 
| 9 | 8 | eqeq2d 2748 | . . . . . 6
⊢ (𝑦 = 𝑥 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}))) | 
| 10 | 9 | cbvrexvw 3238 | . . . . 5
⊢
(∃𝑦 ∈
𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)}) ↔ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})) | 
| 11 | 10 | abbii 2809 | . . . 4
⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} | 
| 12 | 11 | fundcmpsurbijinjpreimafv 47394 | . . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 13 |  | foeq3 6818 | . . . . . . . . 9
⊢ (𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})})) | 
| 14 | 13 | adantl 481 | . . . . . . . 8
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (𝑔:𝐴–onto→𝑝 ↔ 𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})})) | 
| 15 |  | f1oeq23 6839 | . . . . . . . . 9
⊢ ((𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ 𝑞 = (𝐹 “ 𝐴)) → (ℎ:𝑝–1-1-onto→𝑞 ↔ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴))) | 
| 16 | 15 | ancoms 458 | . . . . . . . 8
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (ℎ:𝑝–1-1-onto→𝑞 ↔ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴))) | 
| 17 |  | f1eq2 6800 | . . . . . . . . 9
⊢ (𝑞 = (𝐹 “ 𝐴) → (𝑖:𝑞–1-1→𝐵 ↔ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵)) | 
| 18 | 17 | adantr 480 | . . . . . . . 8
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (𝑖:𝑞–1-1→𝐵 ↔ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵)) | 
| 19 | 14, 16, 18 | 3anbi123d 1438 | . . . . . . 7
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → ((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ↔ (𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵))) | 
| 20 | 19 | anbi1d 631 | . . . . . 6
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))) | 
| 21 | 20 | 3exbidv 1925 | . . . . 5
⊢ ((𝑞 = (𝐹 “ 𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}) → (∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))) | 
| 22 | 21 | spc2egv 3599 | . . . 4
⊢ (((𝐹 “ 𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) → (∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))) | 
| 23 | 22 | imp 406 | . . 3
⊢ ((((𝐹 “ 𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∈ V) ∧ ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})} ∧ ℎ:{𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑦)})}–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 24 | 3, 5, 12, 23 | syl21anc 838 | . 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 25 |  | exrot4 2166 | . . 3
⊢
(∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 26 |  | excom13 2164 | . . . 4
⊢
(∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 27 | 26 | 2exbii 1849 | . . 3
⊢
(∃𝑔∃ℎ∃𝑞∃𝑝∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 28 | 25, 27 | bitri 275 | . 2
⊢
(∃𝑞∃𝑝∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)) ↔ ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) | 
| 29 | 24, 28 | sylib 218 | 1
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔))) |