Theorem funco 5391

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 5026	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 5366	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1923	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4	3	ralrimivw 2616	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦)
5		dffun8 5379	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
6	5	simprbi 275	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
7	4, 6	anim12ci 339	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
8		r19.26 2669	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
9	7, 8	sylibr 134	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥𝐺𝑧
11	10	euexex 2166	. . . . . . 7 ⊢ ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
12	11	ralimi 2605	. . . . . 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 3301	. . . . 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 4109	. . . . . . 7 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4757	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
18	17	eleq2i 2299	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
19		opabid 4373	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
20	16, 18, 19	3bitri 206	. . . . . 6 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
21	20	mobii 2117	. . . . 5 ⊢ (∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
22	21	ralbii 2548	. . . 4 ⊢ (∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
23	15, 22	sylibr 134	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦)
24		relco 5260	. . 3 ⊢ Rel (𝐹 ∘ 𝐺)
25	23, 24	jctil 312	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
26		dffun7 5378	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
27	25, 26	sylibr 134	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396  ∃wex 1541  ∃!weu 2080  ∃*wmo 2081   ∈ wcel 2203  ∀wral 2520   ⊆ wss 3210  ⟨cop 3691   class class class wbr 4108  {copab 4169  dom cdm 4748   ∘ ccom 4752  Rel wrel 4753  Fun wfun 5345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-fun 5353

This theorem is referenced by:  fnco  5465  f1co  5584  fncofn  5861  suppcofn  6465  tposfun  6490  fsuppcorn  7253  casefun  7375  caseinj  7379  caseinl  7381  caseinr  7382  djufun  7394  djuinj  7396  ctssdccl  7401  lidlmex  14615