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| Mirrors > Home > MPE Home > Th. List > f1cof1 | Structured version Visualization version GIF version | ||
| Description: Composition of two one-to-one functions. Generalization of f1co 6815. (Contributed by AV, 18-Sep-2024.) |
| Ref | Expression |
|---|---|
| f1cof1 | ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-f1 6566 | . . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹)) | |
| 2 | df-f1 6566 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
| 3 | ffun 6739 | . . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺) | |
| 4 | fcof 6759 | . . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) | |
| 5 | 3, 4 | sylan2 593 | . . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) |
| 6 | funco 6606 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
| 7 | cnvco 5896 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
| 8 | 7 | funeqi 6587 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
| 9 | 6, 8 | sylibr 234 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
| 10 | 9 | ancoms 458 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
| 11 | 5, 10 | anim12i 613 | . . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
| 12 | 11 | an4s 660 | . . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
| 13 | 1, 2, 12 | syl2anb 598 | . 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
| 14 | df-f1 6566 | . 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) | |
| 15 | 13, 14 | sylibr 234 | 1 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ◡ccnv 5684 “ cima 5688 ∘ ccom 5689 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 |
| This theorem is referenced by: f1co 6815 f1cof1b 47089 |
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