Theorem fvun1 5623

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

Step	Hyp	Ref	Expression
1		fnfun 5351	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1020	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 5351	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1021	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 5353	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 5353	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 3362	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2202	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
9	8	biimprd 158	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
10	9	adantrd 279	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11	10	3impia 1202	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		simp3r 1028	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
13	5	eleq2d 2263	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
14	13	3ad2ant1 1020	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
15	12, 14	mpbird 167	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom 𝐹)
16		funun 5298	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
17		ssun1 3322	. . . . . . . . 9 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
18		dmss 4861	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺))
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺)
20	19	sseli 3175	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ∪ 𝐺))
21	16, 20	anim12i 338	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑋 ∈ dom 𝐹) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
22	21	anasss 399	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
23	22	3impa 1196	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
24		funfvdm 5620	. . . 4 ⊢ ((Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)) → ((𝐹 ∪ 𝐺)‘𝑋) = ∪ ((𝐹 ∪ 𝐺) “ {𝑋}))
25	23, 24	syl 14	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = ∪ ((𝐹 ∪ 𝐺) “ {𝑋}))
26		imaundir 5079	. . . . . 6 ⊢ ((𝐹 ∪ 𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋}))
27	26	a1i 9	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
28	27	unieqd 3846	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29		disjel 3501	. . . . . . . . 9 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30		ndmima 5042	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32	31	3ad2ant3 1022	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
33	32	uneq2d 3313	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34		un0 3480	. . . . . 6 ⊢ ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
35	33, 34	eqtrdi 2242	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
36	35	unieqd 3846	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ∪ (𝐹 “ {𝑋}))
37	28, 36	eqtrd 2226	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ (𝐹 “ {𝑋}))
38		funfvdm 5620	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = ∪ (𝐹 “ {𝑋}))
39	38	eqcomd 2199	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
40	39	adantrl 478	. . . 4 ⊢ ((Fun 𝐹 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
41	40	3adant2 1018	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
42	25, 37, 41	3eqtrd 2230	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
43	2, 4, 11, 15, 42	syl112anc 1253	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164   ∪ cun 3151   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  {csn 3618  ∪ cuni 3835  dom cdm 4659   “ cima 4662  Fun wfun 5248   Fn wfn 5249  ‘cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  fvun2  5624  caseinl  7150