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Theorem fnun 5189
 Description: The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)
Assertion
Ref Expression
fnun (((F Fn A G Fn B) (AB) = ) → (FG) Fn (AB))

Proof of Theorem fnun
StepHypRef Expression
1 df-fn 4790 . . 3 (F Fn A ↔ (Fun F dom F = A))
2 df-fn 4790 . . 3 (G Fn B ↔ (Fun G dom G = B))
3 ineq12 3452 . . . . . . . . . . 11 ((dom F = A dom G = B) → (dom F ∩ dom G) = (AB))
43eqeq1d 2361 . . . . . . . . . 10 ((dom F = A dom G = B) → ((dom F ∩ dom G) = ↔ (AB) = ))
54anbi2d 684 . . . . . . . . 9 ((dom F = A dom G = B) → (((Fun F Fun G) (dom F ∩ dom G) = ) ↔ ((Fun F Fun G) (AB) = )))
6 funun 5146 . . . . . . . . 9 (((Fun F Fun G) (dom F ∩ dom G) = ) → Fun (FG))
75, 6syl6bir 220 . . . . . . . 8 ((dom F = A dom G = B) → (((Fun F Fun G) (AB) = ) → Fun (FG)))
8 dmun 4912 . . . . . . . . 9 dom (FG) = (dom F ∪ dom G)
9 uneq12 3413 . . . . . . . . 9 ((dom F = A dom G = B) → (dom F ∪ dom G) = (AB))
108, 9syl5eq 2397 . . . . . . . 8 ((dom F = A dom G = B) → dom (FG) = (AB))
117, 10jctird 528 . . . . . . 7 ((dom F = A dom G = B) → (((Fun F Fun G) (AB) = ) → (Fun (FG) dom (FG) = (AB))))
12 df-fn 4790 . . . . . . 7 ((FG) Fn (AB) ↔ (Fun (FG) dom (FG) = (AB)))
1311, 12syl6ibr 218 . . . . . 6 ((dom F = A dom G = B) → (((Fun F Fun G) (AB) = ) → (FG) Fn (AB)))
1413exp3a 425 . . . . 5 ((dom F = A dom G = B) → ((Fun F Fun G) → ((AB) = → (FG) Fn (AB))))
1514impcom 419 . . . 4 (((Fun F Fun G) (dom F = A dom G = B)) → ((AB) = → (FG) Fn (AB)))
1615an4s 799 . . 3 (((Fun F dom F = A) (Fun G dom G = B)) → ((AB) = → (FG) Fn (AB)))
171, 2, 16syl2anb 465 . 2 ((F Fn A G Fn B) → ((AB) = → (FG) Fn (AB)))
1817imp 418 1 (((F Fn A G Fn B) (AB) = ) → (FG) Fn (AB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  dom cdm 4772  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:  fnunsn  5190  fun  5236  f1oun  5304  fnfullfun  5858
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