Theorem fvun1 6933

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 6600	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 6600	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1135	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 6603	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 6603	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 4176	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2739	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
9	8	biimprd 248	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
10	9	adantrd 491	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11	10	3impia 1118	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		fvun 6932	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋)))
13	2, 4, 11, 12	syl21anc 838	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋)))
14		disjel 4411	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ 𝐵)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ 𝐵)
16	6	eleq2d 2823	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐵))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐵))
18	15, 17	mtbird 325	. . . . . 6 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
19	18	3adant1 1131	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20		ndmfv 6874	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅)
21	19, 20	syl 17	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) = ∅)
22	21	uneq2d 4122	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = ((𝐹‘𝑋) ∪ ∅))
23		un0 4348	. . 3 ⊢ ((𝐹‘𝑋) ∪ ∅) = (𝐹‘𝑋)
24	22, 23	eqtrdi 2788	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = (𝐹‘𝑋))
25	13, 24	eqtrd 2772	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∪ cun 3901   ∩ cin 3902  ∅c0 4287  dom cdm 5632  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  fvun2  6934  fvun1d  6935  frrlem12  8249  enfixsn  9026  ptunhmeo  23764  noextenddif  27648  axlowdimlem6  29032  axlowdimlem8  29034  axlowdimlem11  29037  vtxdun  29567  isoun  32791  cycpmfv3  33208  lbsdiflsp0  33803  sseqfv1  34566  reprsuc  34792  breprexplema  34807  cvmliftlem5  35502  fullfunfv  36160  finixpnum  37850  poimirlem1  37866  poimirlem2  37867  poimirlem3  37868  poimirlem4  37869  poimirlem6  37871  poimirlem7  37872  poimirlem11  37876  poimirlem12  37877  poimirlem16  37881  poimirlem17  37882  poimirlem19  37884  poimirlem22  37887  poimirlem23  37888  poimirlem28  37893  aacllem  50154