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Mirrors > Home > MPE Home > Th. List > f1co | Structured version Visualization version GIF version |
Description: Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.) |
Ref | Expression |
---|---|
f1co | ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1cof1 6798 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶) | |
2 | f1f 6787 | . . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵) | |
3 | fimacnv 6739 | . . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (◡𝐺 “ 𝐵) = 𝐴) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (◡𝐺 “ 𝐵) = 𝐴) |
6 | 5 | eqcomd 2737 | . . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (◡𝐺 “ 𝐵)) |
7 | f1eq2 6783 | . . 3 ⊢ (𝐴 = (◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶)) |
9 | 1, 8 | mpbird 257 | 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 ↔ wb 205 ∧ wa 395 = wceq 1540 ◡ccnv 5675 “ cima 5679 ∘ ccom 5680 ⟶wf 6539 –1-1→wf1 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 |
This theorem is referenced by: f1oco 6856 f1cofveqaeqALT 7261 tposf12 8240 domtr 9007 domtrfil 9199 dfac12lem2 10143 fin23lem28 10339 pwfseqlem5 10662 cofth 17891 injsubmefmnd 18815 gsumzf1o 19822 cycpmconjv 32572 erdsze2lem2 34494 fcoresf1b 46079 fundcmpsurinjpreimafv 46375 |
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