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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 6792. (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 6541 | . . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹)) | |
2 | df-f1 6541 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | ffun 6713 | . . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺) | |
4 | fcof 6733 | . . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) | |
5 | 3, 4 | sylan2 592 | . . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) |
6 | funco 6581 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
7 | cnvco 5878 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
8 | 7 | funeqi 6562 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
9 | 6, 8 | sylibr 233 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
10 | 9 | ancoms 458 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
11 | 5, 10 | anim12i 612 | . . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
12 | 11 | an4s 657 | . . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
13 | 1, 2, 12 | syl2anb 597 | . 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
14 | df-f1 6541 | . 2 ⊢ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷 ↔ ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) | |
15 | 13, 14 | sylibr 233 | 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 5668 “ cima 5672 ∘ ccom 5673 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 |
This theorem is referenced by: f1co 6792 f1cof1b 46338 |
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