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 2767	. . 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 5650   “ cima 5654   ∘ ccom 5655  ⟶wf 6534  –1-1→wf1 6535

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543

This theorem is used by:  f1oco  6848  f1cofveqaeqALT  7262  tposf12  8268  domtr  9034  domtrfil  9207  dfac12lem2  10223  fin23lem28  10418  pwfseqlem5  10748  cofth  18112  injsubmefmnd  19093  gsumzf1o  20126  cycpmconjv  33703  erdsze2lem2  35969  fcoresf1b  48139  fundcmpsurinjpreimafv  48489