Theorem fvun1 5534

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 5267	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1003	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 5267	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1004	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 5269	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 5269	. . . . . . 7 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 3310	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2166	. . . . 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 1182	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		simp3r 1011	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ 𝐴)
13	5	eleq2d 2227	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴))
14	13	3ad2ant1 1003	. . 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 5214	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
17		ssun1 3270	. . . . . . . . 9 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
18		dmss 4785	. . . . . . . . 9 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺))
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom 𝐹 ⊆ dom (𝐹 ∪ 𝐺)
20	19	sseli 3124	. . . . . . 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 1177	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹 ∪ 𝐺) ∧ 𝑋 ∈ dom (𝐹 ∪ 𝐺)))
24		funfvdm 5531	. . . 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 4999	. . . . . 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 3783	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29		disjel 3448	. . . . . . . . 9 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30		ndmima 4963	. . . . . . . . 9 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
31	29, 30	syl 14	. . . . . . . 8 ⊢ (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32	31	3ad2ant3 1005	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
33	32	uneq2d 3261	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34		un0 3427	. . . . . 6 ⊢ ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
35	33, 34	eqtrdi 2206	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
36	35	unieqd 3783	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ∪ (𝐹 “ {𝑋}))
37	28, 36	eqtrd 2190	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ ((𝐹 ∪ 𝐺) “ {𝑋}) = ∪ (𝐹 “ {𝑋}))
38		funfvdm 5531	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = ∪ (𝐹 “ {𝑋}))
39	38	eqcomd 2163	. . . . 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 1001	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ∪ (𝐹 “ {𝑋}) = (𝐹‘𝑋))
42	25, 37, 41	3eqtrd 2194	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
43	2, 4, 11, 15, 42	syl112anc 1224	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 963   = wceq 1335   ∈ wcel 2128   ∪ cun 3100   ∩ cin 3101   ⊆ wss 3102  ∅c0 3394  {csn 3560  ∪ cuni 3772  dom cdm 4586   “ cima 4589  Fun wfun 5164   Fn wfn 5165  ‘cfv 5170

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178

This theorem is referenced by:  fvun2  5535  caseinl  7035