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Theorem fnco 5191
 Description: Composition of two functions. (Contributed by set.mm contributors, 22-May-2006.)
Assertion
Ref Expression
fnco ((F Fn A G Fn B ran G A) → (F G) Fn B)

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 5181 . . . 4 (F Fn A → Fun F)
2 fnfun 5181 . . . 4 (G Fn B → Fun G)
3 funco 5142 . . . 4 ((Fun F Fun G) → Fun (F G))
41, 2, 3syl2an 463 . . 3 ((F Fn A G Fn B) → Fun (F G))
543adant3 975 . 2 ((F Fn A G Fn B ran G A) → Fun (F G))
6 fndm 5182 . . . . . . 7 (F Fn A → dom F = A)
76sseq2d 3299 . . . . . 6 (F Fn A → (ran G dom F ↔ ran G A))
87biimpar 471 . . . . 5 ((F Fn A ran G A) → ran G dom F)
9 dmcosseq 4973 . . . . 5 (ran G dom F → dom (F G) = dom G)
108, 9syl 15 . . . 4 ((F Fn A ran G A) → dom (F G) = dom G)
11103adant2 974 . . 3 ((F Fn A G Fn B ran G A) → dom (F G) = dom G)
12 fndm 5182 . . . 4 (G Fn B → dom G = B)
13123ad2ant2 977 . . 3 ((F Fn A G Fn B ran G A) → dom G = B)
1411, 13eqtrd 2385 . 2 ((F Fn A G Fn B ran G A) → dom (F G) = B)
15 df-fn 4790 . 2 ((F G) Fn B ↔ (Fun (F G) dom (F G) = B))
165, 14, 15sylanbrc 645 1 ((F Fn A G Fn B ran G A) → (F G) Fn B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ⊆ wss 3257   ∘ ccom 4721  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn wfn 4776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 This theorem is referenced by:  fco  5231  fnfco  5237  xpassen  6057
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