Theorem f1co 6791

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 6790	. 2 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)–1-1→𝐶)
2		f1f 6778	. . . . . 6 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6732	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
4	2, 3	syl 18	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
5	4	adantl 487	. . . 4 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (◡𝐺 “ 𝐵) = 𝐴)
6	5	eqcomd 2771	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → 𝐴 = (◡𝐺 “ 𝐵))
7		f1eq2 6774	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5662   “ cima 5666   ∘ ccom 5667  ⟶wf 6536  –1-1→wf1 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545

This theorem is used by:  f1oco  6848  f1cofveqaeqALT  7261  tposf12  8253  domtr  9010  domtrfil  9183  dfac12lem2  10144  fin23lem28  10339  pwfseqlem5  10663  cofth  18016  injsubmefmnd  18993  gsumzf1o  20026  cycpmconjv  33526  erdsze2lem2  35733  fcoresf1b  47865  fundcmpsurinjpreimafv  48215