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Mirrors > Home > MPE Home > Th. List > f1coOLD | Structured version Visualization version GIF version |
Description: Obsolete version of f1co 6810 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 6558 | . . 3 ⊢ (𝐹:𝐵–1-1→𝐶 ↔ (𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹)) | |
2 | df-f1 6558 | . . 3 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) | |
3 | fco 6752 | . . . . 5 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | |
4 | funco 6598 | . . . . . . 7 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun (◡𝐺 ∘ ◡𝐹)) | |
5 | cnvco 5892 | . . . . . . . 8 ⊢ ◡(𝐹 ∘ 𝐺) = (◡𝐺 ∘ ◡𝐹) | |
6 | 5 | funeqi 6579 | . . . . . . 7 ⊢ (Fun ◡(𝐹 ∘ 𝐺) ↔ Fun (◡𝐺 ∘ ◡𝐹)) |
7 | 4, 6 | sylibr 233 | . . . . . 6 ⊢ ((Fun ◡𝐺 ∧ Fun ◡𝐹) → Fun ◡(𝐹 ∘ 𝐺)) |
8 | 7 | ancoms 457 | . . . . 5 ⊢ ((Fun ◡𝐹 ∧ Fun ◡𝐺) → Fun ◡(𝐹 ∘ 𝐺)) |
9 | 3, 8 | anim12i 611 | . . . 4 ⊢ (((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) ∧ (Fun ◡𝐹 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
10 | 9 | an4s 658 | . . 3 ⊢ (((𝐹:𝐵⟶𝐶 ∧ Fun ◡𝐹) ∧ (𝐺:𝐴⟶𝐵 ∧ Fun ◡𝐺)) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
11 | 1, 2, 10 | syl2anb 596 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) |
12 | df-f1 6558 | . 2 ⊢ ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ∧ Fun ◡(𝐹 ∘ 𝐺))) | |
13 | 11, 12 | sylibr 233 | 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 394 ◡ccnv 5681 ∘ ccom 5686 Fun wfun 6547 ⟶wf 6549 –1-1→wf1 6550 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 |
This theorem is referenced by: (None) |
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