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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 6751. (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 6502 | . . 3 ⊢ (𝐹:𝐶–1-1→𝐷 ↔ (𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹)) | |
2 | df-f1 6502 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | ffun 6672 | . . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → Fun 𝐺) | |
4 | fcof 6692 | . . . . . 6 ⊢ ((𝐹:𝐶⟶𝐷 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) | |
5 | 3, 4 | sylan2 594 | . . . . 5 ⊢ ((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷) |
6 | funco 6542 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
7 | cnvco 5842 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
8 | 7 | funeqi 6523 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
9 | 6, 8 | sylibr 233 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
10 | 9 | ancoms 460 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
11 | 5, 10 | anim12i 614 | . . . 4 ⊢ (((𝐹:𝐶⟶𝐷 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
12 | 11 | an4s 659 | . . 3 ⊢ (((𝐹:𝐶⟶𝐷 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
13 | 1, 2, 12 | syl2anb 599 | . 2 ⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)⟶𝐷 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
14 | df-f1 6502 | . 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 397 ◡ccnv 5633 “ cima 5637 ∘ ccom 5638 Fun wfun 6491 ⟶wf 6493 –1-1→wf1 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 |
This theorem is referenced by: f1co 6751 f1cof1b 45316 |
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