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Theorem fun 5236
 Description: The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)
Assertion
Ref Expression
fun (((F:A–→C G:B–→D) (AB) = ) → (FG):(AB)–→(CD))

Proof of Theorem fun
StepHypRef Expression
1 fnun 5189 . . . . 5 (((F Fn A G Fn B) (AB) = ) → (FG) Fn (AB))
21expcom 424 . . . 4 ((AB) = → ((F Fn A G Fn B) → (FG) Fn (AB)))
3 rnun 5036 . . . . . 6 ran (FG) = (ran F ∪ ran G)
4 unss12 3435 . . . . . 6 ((ran F C ran G D) → (ran F ∪ ran G) (CD))
53, 4syl5eqss 3315 . . . . 5 ((ran F C ran G D) → ran (FG) (CD))
65a1i 10 . . . 4 ((AB) = → ((ran F C ran G D) → ran (FG) (CD)))
72, 6anim12d 546 . . 3 ((AB) = → (((F Fn A G Fn B) (ran F C ran G D)) → ((FG) Fn (AB) ran (FG) (CD))))
8 df-f 4791 . . . . 5 (F:A–→C ↔ (F Fn A ran F C))
9 df-f 4791 . . . . 5 (G:B–→D ↔ (G Fn B ran G D))
108, 9anbi12i 678 . . . 4 ((F:A–→C G:B–→D) ↔ ((F Fn A ran F C) (G Fn B ran G D)))
11 an4 797 . . . 4 (((F Fn A ran F C) (G Fn B ran G D)) ↔ ((F Fn A G Fn B) (ran F C ran G D)))
1210, 11bitri 240 . . 3 ((F:A–→C G:B–→D) ↔ ((F Fn A G Fn B) (ran F C ran G D)))
13 df-f 4791 . . 3 ((FG):(AB)–→(CD) ↔ ((FG) Fn (AB) ran (FG) (CD)))
147, 12, 133imtr4g 261 . 2 ((AB) = → ((F:A–→C G:B–→D) → (FG):(AB)–→(CD)))
1514impcom 419 1 (((F:A–→C G:B–→D) (AB) = ) → (FG):(AB)–→(CD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  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: (None)
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