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) ∧ (A ∩ B) = ∅) → (F ∪ G):(A ∪ B)–→(C ∪ D))

Proof of Theorem fun

Step	Hyp	Ref	Expression
1		fnun 5189	. . . . 5 ⊢ (((F Fn A ∧ G Fn B) ∧ (A ∩ B) = ∅) → (F ∪ G) Fn (A ∪ B))
2	1	expcom 424	. . . 4 ⊢ ((A ∩ B) = ∅ → ((F Fn A ∧ G Fn B) → (F ∪ G) Fn (A ∪ B)))
3		rnun 5036	. . . . . 6 ⊢ ran (F ∪ G) = (ran F ∪ ran G)
4		unss12 3435	. . . . . 6 ⊢ ((ran F ⊆ C ∧ ran G ⊆ D) → (ran F ∪ ran G) ⊆ (C ∪ D))
5	3, 4	syl5eqss 3315	. . . . 5 ⊢ ((ran F ⊆ C ∧ ran G ⊆ D) → ran (F ∪ G) ⊆ (C ∪ D))
6	5	a1i 10	. . . 4 ⊢ ((A ∩ B) = ∅ → ((ran F ⊆ C ∧ ran G ⊆ D) → ran (F ∪ G) ⊆ (C ∪ D)))
7	2, 6	anim12d 546	. . 3 ⊢ ((A ∩ B) = ∅ → (((F Fn A ∧ G Fn B) ∧ (ran F ⊆ C ∧ ran G ⊆ D)) → ((F ∪ G) Fn (A ∪ B) ∧ ran (F ∪ G) ⊆ (C ∪ D))))
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))
10	8, 9	anbi12i 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)))
12	10, 11	bitri 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 ⊢ ((F ∪ G):(A ∪ B)–→(C ∪ D) ↔ ((F ∪ G) Fn (A ∪ B) ∧ ran (F ∪ G) ⊆ (C ∪ D)))
14	7, 12, 13	3imtr4g 261	. 2 ⊢ ((A ∩ B) = ∅ → ((F:A–→C ∧ G:B–→D) → (F ∪ G):(A ∪ B)–→(C ∪ D)))
15	14	impcom 419	1 ⊢ (((F:A–→C ∧ G:B–→D) ∧ (A ∩ B) = ∅) → (F ∪ G):(A ∪ B)–→(C ∪ D))

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-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  df-f 4791

This theorem is referenced by: (None)