Theorem funco 5143

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		funmo 5126	. . . . 5 ⊢ (Fun G → ∃*z xGz)
2		funmo 5126	. . . . . 6 ⊢ (Fun F → ∃*y zFy)
3	2	alrimiv 1631	. . . . 5 ⊢ (Fun F → ∀z∃*y zFy)
4		moexexv 2274	. . . . 5 ⊢ ((∃*z xGz ∧ ∀z∃*y zFy) → ∃*y∃z(xGz ∧ zFy))
5	1, 3, 4	syl2anr 464	. . . 4 ⊢ ((Fun F ∧ Fun G) → ∃*y∃z(xGz ∧ zFy))
6	5	alrimiv 1631	. . 3 ⊢ ((Fun F ∧ Fun G) → ∀x∃*y∃z(xGz ∧ zFy))
7		funopab 5140	. . 3 ⊢ (Fun {⟨x, y⟩ ∣ ∃z(xGz ∧ zFy)} ↔ ∀x∃*y∃z(xGz ∧ zFy))
8	6, 7	sylibr 203	. 2 ⊢ ((Fun F ∧ Fun G) → Fun {⟨x, y⟩ ∣ ∃z(xGz ∧ zFy)})
9		df-co 4727	. . 3 ⊢ (F ∘ G) = {⟨x, y⟩ ∣ ∃z(xGz ∧ zFy)}
10	9	funeqi 5129	. 2 ⊢ (Fun (F ∘ G) ↔ Fun {⟨x, y⟩ ∣ ∃z(xGz ∧ zFy)})
11	8, 10	sylibr 203	1 ⊢ ((Fun F ∧ Fun G) → Fun (F ∘ G))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃*wmo 2205  {copab 4623   class class class wbr 4640   ∘ ccom 4722  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by:  fnco  5192  f1co  5265  sbthlem3  6206