Proof of Theorem f1cof1b
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simp1 1137 | . . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶𝐵) | 
| 2 |  | eqid 2737 | . . . . . . . . 9
⊢ (ran
𝐹 ∩ 𝐶) = (ran 𝐹 ∩ 𝐶) | 
| 3 |  | eqid 2737 | . . . . . . . . 9
⊢ (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶) | 
| 4 |  | eqid 2737 | . . . . . . . . 9
⊢ (𝐹 ↾ (◡𝐹 “ 𝐶)) = (𝐹 ↾ (◡𝐹 “ 𝐶)) | 
| 5 |  | simp2 1138 | . . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐺:𝐶⟶𝐷) | 
| 6 |  | eqid 2737 | . . . . . . . . 9
⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) | 
| 7 |  | simp3 1139 | . . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶) | 
| 8 | 1, 2, 3, 4, 5, 6, 7 | f1cof1blem 47086 | . . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺))) | 
| 9 |  | simpll 767 | . . . . . . . 8
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)) → (◡𝐹 “ 𝐶) = 𝐴) | 
| 10 |  | f1eq2 6800 | . . . . . . . 8
⊢ ((◡𝐹 “ 𝐶) = 𝐴 → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷)) | 
| 11 | 8, 9, 10 | 3syl 18 | . . . . . . 7
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷)) | 
| 12 | 11 | bicomd 223 | . . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷)) | 
| 13 |  | ancom 460 | . . . . . . . . . 10
⊢ (((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹) ↔ ((𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺)) | 
| 14 | 13 | anbi2i 623 | . . . . . . . . 9
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) ↔ (((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺))) | 
| 15 | 8, 14 | sylibr 234 | . . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹))) | 
| 16 | 15 | adantr 480 | . . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → (((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹))) | 
| 17 | 1 | adantr 480 | . . . . . . . . 9
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → 𝐹:𝐴⟶𝐵) | 
| 18 | 5 | adantr 480 | . . . . . . . . 9
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → 𝐺:𝐶⟶𝐷) | 
| 19 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) | 
| 20 | 17, 2, 3, 4, 18, 6,
19 | fcoresf1 47081 | . . . . . . . 8
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → ((𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) ∧ (𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷)) | 
| 21 | 20 | ancomd 461 | . . . . . . 7
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶))) | 
| 22 |  | simprl 771 | . . . . . . . . . 10
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → (𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺) | 
| 23 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) → (ran 𝐹 ∩ 𝐶) = 𝐶) | 
| 24 | 23 | adantr 480 | . . . . . . . . . 10
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → (ran 𝐹 ∩ 𝐶) = 𝐶) | 
| 25 |  | eqidd 2738 | . . . . . . . . . 10
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → 𝐷 = 𝐷) | 
| 26 | 22, 24, 25 | f1eq123d 6840 | . . . . . . . . 9
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ↔ 𝐺:𝐶–1-1→𝐷)) | 
| 27 | 26 | biimpd 229 | . . . . . . . 8
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 → 𝐺:𝐶–1-1→𝐷)) | 
| 28 |  | simprr 773 | . . . . . . . . . 10
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹) | 
| 29 |  | simpll 767 | . . . . . . . . . 10
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → (◡𝐹 “ 𝐶) = 𝐴) | 
| 30 | 28, 29, 24 | f1eq123d 6840 | . . . . . . . . 9
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) ↔ 𝐹:𝐴–1-1→𝐶)) | 
| 31 | 30 | biimpd 229 | . . . . . . . 8
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → ((𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶) → 𝐹:𝐴–1-1→𝐶)) | 
| 32 | 27, 31 | anim12d 609 | . . . . . . 7
⊢ ((((◡𝐹 “ 𝐶) = 𝐴 ∧ (ran 𝐹 ∩ 𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹 ∩ 𝐶)) = 𝐺 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)) = 𝐹)) → (((𝐺 ↾ (ran 𝐹 ∩ 𝐶)):(ran 𝐹 ∩ 𝐶)–1-1→𝐷 ∧ (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1→(ran 𝐹 ∩ 𝐶)) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶))) | 
| 33 | 16, 21, 32 | sylc 65 | . . . . . 6
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶)) | 
| 34 | 12, 33 | sylbida 592 | . . . . 5
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶)) | 
| 35 |  | ffrn 6749 | . . . . . . . . . . . 12
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹) | 
| 36 |  | ax-1 6 | . . . . . . . . . . . 12
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴⟶ran 𝐹 → 𝐹:𝐴⟶𝐵)) | 
| 37 | 35, 36 | impbid2 226 | . . . . . . . . . . 11
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹)) | 
| 38 | 37 | anbi1d 631 | . . . . . . . . . 10
⊢ (𝐹:𝐴⟶𝐵 → ((𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun ◡𝐹))) | 
| 39 |  | df-f1 6566 | . . . . . . . . . 10
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | 
| 40 |  | df-f1 6566 | . . . . . . . . . 10
⊢ (𝐹:𝐴–1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun ◡𝐹)) | 
| 41 | 38, 39, 40 | 3bitr4g 314 | . . . . . . . . 9
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→ran 𝐹)) | 
| 42 | 41 | 3ad2ant1 1134 | . . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→ran 𝐹)) | 
| 43 |  | f1eq3 6801 | . . . . . . . . 9
⊢ (ran
𝐹 = 𝐶 → (𝐹:𝐴–1-1→ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐶)) | 
| 44 | 43 | 3ad2ant3 1136 | . . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→ran 𝐹 ↔ 𝐹:𝐴–1-1→𝐶)) | 
| 45 | 42, 44 | bitrd 279 | . . . . . . 7
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1→𝐶)) | 
| 46 | 45 | anbi2d 630 | . . . . . 6
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶))) | 
| 47 | 46 | adantr 480 | . . . . 5
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) ↔ (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐶))) | 
| 48 | 34, 47 | mpbird 257 | . . . 4
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵)) | 
| 49 | 48 | ancomd 461 | . . 3
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷) → (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)) | 
| 50 | 49 | ex 412 | . 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 → (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷))) | 
| 51 |  | f1cof1 6814 | . . . 4
⊢ ((𝐺:𝐶–1-1→𝐷 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) | 
| 52 | 51 | ancoms 458 | . . 3
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷) | 
| 53 |  | imaeq2 6074 | . . . . . . . 8
⊢ (𝐶 = ran 𝐹 → (◡𝐹 “ 𝐶) = (◡𝐹 “ ran 𝐹)) | 
| 54 |  | cnvimarndm 6101 | . . . . . . . 8
⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | 
| 55 | 53, 54 | eqtrdi 2793 | . . . . . . 7
⊢ (𝐶 = ran 𝐹 → (◡𝐹 “ 𝐶) = dom 𝐹) | 
| 56 | 55 | eqcoms 2745 | . . . . . 6
⊢ (ran
𝐹 = 𝐶 → (◡𝐹 “ 𝐶) = dom 𝐹) | 
| 57 | 56 | 3ad2ant3 1136 | . . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (◡𝐹 “ 𝐶) = dom 𝐹) | 
| 58 | 1 | fdmd 6746 | . . . . 5
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → dom 𝐹 = 𝐴) | 
| 59 | 57, 58 | eqtrd 2777 | . . . 4
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → (◡𝐹 “ 𝐶) = 𝐴) | 
| 60 | 59, 10 | syl 17 | . . 3
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1→𝐷 ↔ (𝐺 ∘ 𝐹):𝐴–1-1→𝐷)) | 
| 61 | 52, 60 | imbitrid 244 | . 2
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) → (𝐺 ∘ 𝐹):𝐴–1-1→𝐷)) | 
| 62 | 50, 61 | impbid 212 | 1
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷))) |