New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  foco GIF version

Theorem foco 5279
 Description: Composition of onto functions. (Contributed by set.mm contributors, 22-Mar-2006.)
Assertion
Ref Expression
foco ((F:BontoC G:AontoB) → (F G):AontoC)

Proof of Theorem foco
StepHypRef Expression
1 fco 5231 . . . 4 ((F:B–→C G:A–→B) → (F G):A–→C)
21ad2ant2r 727 . . 3 (((F:B–→C ran F = C) (G:A–→B ran G = B)) → (F G):A–→C)
3 fdm 5226 . . . . . . 7 (F:B–→C → dom F = B)
4 eqtr3 2372 . . . . . . 7 ((dom F = B ran G = B) → dom F = ran G)
53, 4sylan 457 . . . . . 6 ((F:B–→C ran G = B) → dom F = ran G)
6 rncoeq 4975 . . . . . . . 8 (dom F = ran G → ran (F G) = ran F)
76eqeq1d 2361 . . . . . . 7 (dom F = ran G → (ran (F G) = C ↔ ran F = C))
87biimpar 471 . . . . . 6 ((dom F = ran G ran F = C) → ran (F G) = C)
95, 8sylan 457 . . . . 5 (((F:B–→C ran G = B) ran F = C) → ran (F G) = C)
109an32s 779 . . . 4 (((F:B–→C ran F = C) ran G = B) → ran (F G) = C)
1110adantrl 696 . . 3 (((F:B–→C ran F = C) (G:A–→B ran G = B)) → ran (F G) = C)
122, 11jca 518 . 2 (((F:B–→C ran F = C) (G:A–→B ran G = B)) → ((F G):A–→C ran (F G) = C))
13 dffo2 5273 . . 3 (F:BontoC ↔ (F:B–→C ran F = C))
14 dffo2 5273 . . 3 (G:AontoB ↔ (G:A–→B ran G = B))
1513, 14anbi12i 678 . 2 ((F:BontoC G:AontoB) ↔ ((F:B–→C ran F = C) (G:A–→B ran G = B)))
16 dffo2 5273 . 2 ((F G):AontoC ↔ ((F G):A–→C ran (F G) = C))
1712, 15, 163imtr4i 257 1 ((F:BontoC G:AontoB) → (F G):AontoC)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∘ ccom 4721  dom cdm 4772  ran crn 4773  –→wf 4777  –onto→wfo 4779 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  df-fo 4793 This theorem is referenced by:  f1oco  5308
 Copyright terms: Public domain W3C validator