Theorem fvun1 5745

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 5455	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1045	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 5455	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1046	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 5457	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 5457	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 3426	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2243	. . . . 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 1227	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		simp3r 1053	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
13	5	eleq2d 2304	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
14	13	3ad2ant1 1045	. . 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 5399	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
17		ssun1 3384	. . . . . . . . 9 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
18		dmss 4957	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺))
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺)
20	19	sseli 3236	. . . . . . 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 1221	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
24		funfvdm 5742	. . . 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 5178	. . . . . 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 3927	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29		disjel 3565	. . . . . . . . 9 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30		ndmima 5141	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32	31	3ad2ant3 1047	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
33	32	uneq2d 3375	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34		un0 3544	. . . . . 6 ⊢ ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
35	33, 34	eqtrdi 2283	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
36	35	unieqd 3927	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ∪ (𝐹 “ {𝑋}))
37	28, 36	eqtrd 2267	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ (𝐹 “ {𝑋}))
38		funfvdm 5742	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = ∪ (𝐹 “ {𝑋}))
39	38	eqcomd 2240	. . . . 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 1043	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
42	25, 37, 41	3eqtrd 2271	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
43	2, 4, 11, 15, 42	syl112anc 1278	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 1005   = wceq 1398   ∈ wcel 2205   ∪ cun 3211   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510  {csn 3691  ∪ cuni 3916  dom cdm 4751   “ cima 4754  Fun wfun 5348   Fn wfn 5349  ‘cfv 5354

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  fvun2  5746  caseinl  7384  vtxdfifiun  16341