Theorem fvun1 5552

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 5285	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1008	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 5285	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1009	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 5287	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 5287	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 3325	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2174	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
9	8	biimprd 157	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
10	9	adantrd 277	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11	10	3impia 1190	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		simp3r 1016	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
13	5	eleq2d 2236	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
14	13	3ad2ant1 1008	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
15	12, 14	mpbird 166	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom 𝐹)
16		funun 5232	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
17		ssun1 3285	. . . . . . . . 9 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
18		dmss 4803	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺))
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺)
20	19	sseli 3138	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ∪ 𝐺))
21	16, 20	anim12i 336	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑋 ∈ dom 𝐹) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
22	21	anasss 397	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
23	22	3impa 1184	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
24		funfvdm 5549	. . . 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 5017	. . . . . 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 3800	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29		disjel 3463	. . . . . . . . 9 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30		ndmima 4981	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32	31	3ad2ant3 1010	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
33	32	uneq2d 3276	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34		un0 3442	. . . . . 6 ⊢ ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
35	33, 34	eqtrdi 2215	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
36	35	unieqd 3800	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ∪ (𝐹 “ {𝑋}))
37	28, 36	eqtrd 2198	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ (𝐹 “ {𝑋}))
38		funfvdm 5549	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = ∪ (𝐹 “ {𝑋}))
39	38	eqcomd 2171	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
40	39	adantrl 470	. . . 4 ⊢ ((Fun 𝐹 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
41	40	3adant2 1006	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
42	25, 37, 41	3eqtrd 2202	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
43	2, 4, 11, 15, 42	syl112anc 1232	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  {csn 3576  ∪ cuni 3789  dom cdm 4604   “ cima 4607  Fun wfun 5182   Fn wfn 5183  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  fvun2  5553  caseinl  7056