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Mirrors > Home > MPE Home > Th. List > f1coOLD | Structured version Visualization version GIF version |
Description: Obsolete version of f1co 6605 as of 20-Sep-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
f1coOLD | ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f1 6363 | . . 3 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹)) | |
2 | df-f1 6363 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | fco 6547 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | funco 6398 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
5 | cnvco 5739 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
6 | 5 | funeqi 6379 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
7 | 4, 6 | sylibr 237 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
8 | 7 | ancoms 462 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
9 | 3, 8 | anim12i 616 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
10 | 9 | an4s 660 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
11 | 1, 2, 10 | syl2anb 601 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
12 | df-f1 6363 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) | |
13 | 11, 12 | sylibr 237 | 1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ◡ccnv 5535 ∘ ccom 5540 Fun wfun 6352 ⟶wf 6354 –1-1→wf1 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 |
This theorem is referenced by: (None) |
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