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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 6816. (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 6568 | . . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹)) | |
2 | df-f1 6568 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | ffun 6740 | . . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺) | |
4 | fcof 6760 | . . . . . 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 6608 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
7 | cnvco 5899 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
8 | 7 | funeqi 6589 | . . . . . . 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 6568 | . 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 5688 “ cima 5692 ∘ ccom 5693 Fun wfun 6557 ⟶wf 6559 –1-1→wf1 6560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 |
This theorem is referenced by: f1co 6816 f1cof1b 47027 |
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