Theorem funco 5316

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 4953	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 5291	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1898	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4	3	ralrimivw 2581	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦)
5		dffun8 5304	. . . . . . . . 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 2633	. . . . . . 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 1552	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥𝐺𝑧
11	10	euexex 2140	. . . . . . 7 ⊢ ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
12	11	ralimi 2570	. . . . . 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 3258	. . . . 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 4048	. . . . . . 7 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4688	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
18	17	eleq2i 2273	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
19		opabid 4306	. . . . . . 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 2092	. . . . 5 ⊢ (∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
22	21	ralbii 2513	. . . 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 5186	. . 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 5303	. 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 1371  ∃wex 1516  ∃!weu 2055  ∃*wmo 2056   ∈ wcel 2177  ∀wral 2485   ⊆ wss 3167  ⟨cop 3637   class class class wbr 4047  {copab 4108  dom cdm 4679   ∘ ccom 4683  Rel wrel 4684  Fun wfun 5270

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-fun 5278

This theorem is referenced by:  fnco  5389  f1co  5500  tposfun  6353  casefun  7194  caseinj  7198  caseinl  7200  caseinr  7201  djufun  7213  djuinj  7215  ctssdccl  7220  lidlmex  14281