Theorem funco 5256

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 4896	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 5231	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1874	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4	3	ralrimivw 2551	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦)
5		dffun8 5244	. . . . . . . . 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 2603	. . . . . . 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 1528	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥𝐺𝑧
11	10	euexex 2111	. . . . . . 7 ⊢ ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
12	11	ralimi 2540	. . . . . 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 3219	. . . . 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 4004	. . . . . . 7 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4635	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
18	17	eleq2i 2244	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
19		opabid 4257	. . . . . . 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 2063	. . . . 5 ⊢ (∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
22	21	ralbii 2483	. . . 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 5127	. . 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 5243	. 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 1351  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027   ∈ wcel 2148  ∀wral 2455   ⊆ wss 3129  ⟨cop 3595   class class class wbr 4003  {copab 4063  dom cdm 4626   ∘ ccom 4630  Rel wrel 4631  Fun wfun 5210

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-fun 5218

This theorem is referenced by:  fnco  5324  f1co  5433  tposfun  6260  casefun  7083  caseinj  7087  caseinl  7089  caseinr  7090  djufun  7102  djuinj  7104  ctssdccl  7109