Theorem fco 5231

Description: Composition of two mappings. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 29-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

Ref	Expression
fco	⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C)

Proof of Theorem fco

Step	Hyp	Ref	Expression
1		fnco 5191	. . . . . 6 ⊢ ((F Fn B ∧ G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A)
2	1	3expib 1154	. . . . 5 ⊢ (F Fn B → ((G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A))
3	2	adantr 451	. . . 4 ⊢ ((F Fn B ∧ ran F ⊆ C) → ((G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A))
4		rncoss 4972	. . . . . 6 ⊢ ran (F ∘ G) ⊆ ran F
5		sstr 3280	. . . . . 6 ⊢ ((ran (F ∘ G) ⊆ ran F ∧ ran F ⊆ C) → ran (F ∘ G) ⊆ C)
6	4, 5	mpan 651	. . . . 5 ⊢ (ran F ⊆ C → ran (F ∘ G) ⊆ C)
7	6	adantl 452	. . . 4 ⊢ ((F Fn B ∧ ran F ⊆ C) → ran (F ∘ G) ⊆ C)
8	3, 7	jctird 528	. . 3 ⊢ ((F Fn B ∧ ran F ⊆ C) → ((G Fn A ∧ ran G ⊆ B) → ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C)))
9	8	imp 418	. 2 ⊢ (((F Fn B ∧ ran F ⊆ C) ∧ (G Fn A ∧ ran G ⊆ B)) → ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C))
10		df-f 4791	. . 3 ⊢ (F:B–→C ↔ (F Fn B ∧ ran F ⊆ C))
11		df-f 4791	. . 3 ⊢ (G:A–→B ↔ (G Fn A ∧ ran G ⊆ B))
12	10, 11	anbi12i 678	. 2 ⊢ ((F:B–→C ∧ G:A–→B) ↔ ((F Fn B ∧ ran F ⊆ C) ∧ (G Fn A ∧ ran G ⊆ B)))
13		df-f 4791	. 2 ⊢ ((F ∘ G):A–→C ↔ ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C))
14	9, 12, 13	3imtr4i 257	1 ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C)

Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 3257   ∘ ccom 4721  ran crn 4773   Fn wfn 4776  –→wf 4777

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791

This theorem is referenced by:  f1co  5264  foco  5279  enmap2lem5  6067  enmap1lem5  6073