Theorem funco 5158

Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funco	⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Proof of Theorem funco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmcoss 4803	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 5133	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1846	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4	3	ralrimivw 2504	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦)
5		dffun8 5146	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
6	5	simprbi 273	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
7	4, 6	anim12ci 337	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
8		r19.26 2556	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
9	7, 8	sylibr 133	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10		nfv 1508	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥𝐺𝑧
11	10	euexex 2082	. . . . . . 7 ⊢ ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
12	11	ralimi 2493	. . . . . 6 ⊢ (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
13	9, 12	syl 14	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
14		ssralv 3156	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 → (∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)))
15	1, 13, 14	mpsyl 65	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
16		df-br 3925	. . . . . . 7 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4543	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
18	17	eleq2i 2204	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
19		opabid 4174	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
20	16, 18, 19	3bitri 205	. . . . . 6 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
21	20	mobii 2034	. . . . 5 ⊢ (∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
22	21	ralbii 2439	. . . 4 ⊢ (∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
23	15, 22	sylibr 133	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦)
24		relco 5032	. . 3 ⊢ Rel (𝐹 ∘ 𝐺)
25	23, 24	jctil 310	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
26		dffun7 5145	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
27	25, 26	sylibr 133	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329  ∃wex 1468   ∈ wcel 1480  ∃!weu 1997  ∃*wmo 1998  ∀wral 2414   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924  {copab 3983  dom cdm 4534   ∘ ccom 4538  Rel wrel 4539  Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-fun 5120

This theorem is referenced by:  fnco  5226  f1co  5335  tposfun  6150  casefun  6963  caseinj  6967  caseinl  6969  caseinr  6970  djufun  6982  djuinj  6984  ctssdccl  6989