Theorem fvun1 5583

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 5314	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1018	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 5314	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1019	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 5316	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 5316	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 3339	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2186	. . . . 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 1200	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		simp3r 1026	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
13	5	eleq2d 2247	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
14	13	3ad2ant1 1018	. . 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 5261	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
17		ssun1 3299	. . . . . . . . 9 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
18		dmss 4827	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺))
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺)
20	19	sseli 3152	. . . . . . 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 1194	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
24		funfvdm 5580	. . . 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 5043	. . . . . 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 3821	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29		disjel 3478	. . . . . . . . 9 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30		ndmima 5006	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32	31	3ad2ant3 1020	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
33	32	uneq2d 3290	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34		un0 3457	. . . . . 6 ⊢ ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
35	33, 34	eqtrdi 2226	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
36	35	unieqd 3821	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ∪ (𝐹 “ {𝑋}))
37	28, 36	eqtrd 2210	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ (𝐹 “ {𝑋}))
38		funfvdm 5580	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = ∪ (𝐹 “ {𝑋}))
39	38	eqcomd 2183	. . . . 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 1016	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
42	25, 37, 41	3eqtrd 2214	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
43	2, 4, 11, 15, 42	syl112anc 1242	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 978   = wceq 1353   ∈ wcel 2148   ∪ cun 3128   ∩ cin 3129   ⊆ wss 3130  ∅c0 3423  {csn 3593  ∪ cuni 3810  dom cdm 4627   “ cima 4630  Fun wfun 5211   Fn wfn 5212  ‘cfv 5217

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225

This theorem is referenced by:  fvun2  5584  caseinl  7090