Theorem fvun1 5380

Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Assertion

Ref	Expression
fvun1	⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ((F ∪ G) ‘X) = (F ‘X))

Proof of Theorem fvun1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnfun 5182	. . . . 5 ⊢ (F Fn A → Fun F)
2	1	3ad2ant1 976	. . . 4 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → Fun F)
3		fnfun 5182	. . . . 5 ⊢ (G Fn B → Fun G)
4	3	3ad2ant2 977	. . . 4 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → Fun G)
5		fndm 5183	. . . . . . . . 9 ⊢ (F Fn A → dom F = A)
6		fndm 5183	. . . . . . . . 9 ⊢ (G Fn B → dom G = B)
7		ineq12 3453	. . . . . . . . 9 ⊢ ((dom F = A ∧ dom G = B) → (dom F ∩ dom G) = (A ∩ B))
8	5, 6, 7	syl2an 463	. . . . . . . 8 ⊢ ((F Fn A ∧ G Fn B) → (dom F ∩ dom G) = (A ∩ B))
9	8	eqeq1d 2361	. . . . . . 7 ⊢ ((F Fn A ∧ G Fn B) → ((dom F ∩ dom G) = ∅ ↔ (A ∩ B) = ∅))
10	9	biimprd 214	. . . . . 6 ⊢ ((F Fn A ∧ G Fn B) → ((A ∩ B) = ∅ → (dom F ∩ dom G) = ∅))
11	10	adantrd 454	. . . . 5 ⊢ ((F Fn A ∧ G Fn B) → (((A ∩ B) = ∅ ∧ X ∈ A) → (dom F ∩ dom G) = ∅))
12	11	3impia 1148	. . . 4 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → (dom F ∩ dom G) = ∅)
13		fvun 5379	. . . 4 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((F ∪ G) ‘X) = ((F ‘X) ∪ (G ‘X)))
14	2, 4, 12, 13	syl21anc 1181	. . 3 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ((F ∪ G) ‘X) = ((F ‘X) ∪ (G ‘X)))
15		disj 3592	. . . . . . . . 9 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B)
16		eleq1 2413	. . . . . . . . . . 11 ⊢ (x = X → (x ∈ B ↔ X ∈ B))
17	16	notbid 285	. . . . . . . . . 10 ⊢ (x = X → (¬ x ∈ B ↔ ¬ X ∈ B))
18	17	rspccv 2953	. . . . . . . . 9 ⊢ (∀x ∈ A ¬ x ∈ B → (X ∈ A → ¬ X ∈ B))
19	15, 18	sylbi 187	. . . . . . . 8 ⊢ ((A ∩ B) = ∅ → (X ∈ A → ¬ X ∈ B))
20	19	imp 418	. . . . . . 7 ⊢ (((A ∩ B) = ∅ ∧ X ∈ A) → ¬ X ∈ B)
21	20	3ad2ant3 978	. . . . . 6 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ¬ X ∈ B)
22	6	3ad2ant2 977	. . . . . . 7 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → dom G = B)
23	22	eleq2d 2420	. . . . . 6 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → (X ∈ dom G ↔ X ∈ B))
24	21, 23	mtbird 292	. . . . 5 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ¬ X ∈ dom G)
25		ndmfv 5350	. . . . 5 ⊢ (¬ X ∈ dom G → (G ‘X) = ∅)
26	24, 25	syl 15	. . . 4 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → (G ‘X) = ∅)
27	26	uneq2d 3419	. . 3 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ((F ‘X) ∪ (G ‘X)) = ((F ‘X) ∪ ∅))
28	14, 27	eqtrd 2385	. 2 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ((F ∪ G) ‘X) = ((F ‘X) ∪ ∅))
29		un0 3576	. 2 ⊢ ((F ‘X) ∪ ∅) = (F ‘X)
30	28, 29	syl6eq 2401	1 ⊢ ((F Fn A ∧ G Fn B ∧ ((A ∩ B) = ∅ ∧ X ∈ A)) → ((F ∪ G) ‘X) = (F ‘X))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  fvun2  5381  fvfullfun  5865