Theorem f1co 6821

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⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → (𝐹 ∘ 𝐺):𝐴⟶1-1𝐶)

Proof of Theorem f1co

Step	Hyp	Ref	Expression
1		f1cof1 6820	. 2 ⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)⟶1-1𝐶)
2		f1f 6808	. . . . . 6 ⊢ (𝐺:𝐴⟶1-1𝐵 → 𝐺:𝐴⟶𝐵)
3		fimacnv 6762	. . . . . 6 ⊢ (𝐺:𝐴⟶𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
4	2, 3	syl 18	. . . . 5 ⊢ (𝐺:𝐴⟶1-1𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
5	4	adantl 487	. . . 4 ⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → (◡𝐺 “ 𝐵) = 𝐴)
6	5	eqcomd 2769	. . 3 ⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → 𝐴 = (◡𝐺 “ 𝐵))
7		f1eq2 6804	. . 3 ⊢ (𝐴 = (◡𝐺 “ 𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶1-1𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)⟶1-1𝐶))
8	6, 7	syl 18	. 2 ⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → ((𝐹 ∘ 𝐺):𝐴⟶1-1𝐶 ↔ (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐵)⟶1-1𝐶))
9	1, 8	mpbird 260	1 ⊢ ((𝐹:𝐵⟶1-1𝐶 ∧ 𝐺:𝐴⟶1-1𝐵) → (𝐹 ∘ 𝐺):𝐴⟶1-1𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5657   “ cima 5661   ∘ ccom 5662  ⟶wf 6562  ⟶1-1wf1 6563

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571

This theorem is used by:  f1oco  6878  f1cofveqaeqALT  7293  tposf12  8306  domtr  9072  domtrfil  9245  dfac12lem2  10262  fin23lem28  10457  pwfseqlem5  10787  cofth  18162  injsubmefmnd  19143  gsumzf1o  20177  cycpmconjv  33802  onprcf1of1orad  36094  erdsze2lem2  36177  fcoresf1b  48413  fundcmpsurinjpreimafv  48763