Theorem f1co 6787

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.)

Assertion

Ref	Expression
f1co	⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)

Proof of Theorem f1co

Step	Hyp	Ref	Expression
1		f1cof1 6786	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶)
2		f1f 6774	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6728	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
4	2, 3	syl 18	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
5	4	adantl 486	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (◡𝐺 “ 𝐵) = 𝐴)
6	5	eqcomd 2769	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (◡𝐺 “ 𝐵))
7		f1eq2 6770	. . 3 ⊢ (𝐴 = (◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶))
8	6, 7	syl 18	. 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 260	1 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ^◡ccnv 5660   “ cima 5664   ∘ ccom 5665  ⟶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 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541

This theorem is referenced by:  f1oco  6844  f1cofveqaeqALT  7256  tposf12  8243  domtr  9000  domtrfil  9172  dfac12lem2  10124  fin23lem28  10319  pwfseqlem5  10643  cofth  17989  injsubmefmnd  18951  gsumzf1o  19977  cycpmconjv  33462  erdsze2lem2  35696  fcoresf1b  47807  fundcmpsurinjpreimafv  48157