Theorem fvun1 6918

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 6586	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹)
3		fnfun 6586	. . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
4	3	3ad2ant2 1134	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺)
5		fndm 6589	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6		fndm 6589	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
7	5, 6	ineqan12d 4175	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
8	7	eqeq1d 2731	. . . . . 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 1117	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12		fvun 6917	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋)))
13	2, 4, 11, 12	syl21anc 837	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋)))
14		disjel 4410	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ 𝐵)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ 𝐵)
16	6	eleq2d 2814	. . . . . . . 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 1130	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20		ndmfv 6859	. . . . 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 4121	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = ((𝐹‘𝑋) ∪ ∅))
23		un0 4347	. . 3 ⊢ ((𝐹‘𝑋) ∪ ∅) = (𝐹‘𝑋)
24	22, 23	eqtrdi 2780	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = (𝐹‘𝑋))
25	13, 24	eqtrd 2764	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 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3903   ∩ cin 3904  ∅c0 4286  dom cdm 5623  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  fvun2  6919  fvun1d  6920  frrlem12  8237  enfixsn  9010  ptunhmeo  23711  noextenddif  27596  axlowdimlem6  28910  axlowdimlem8  28912  axlowdimlem11  28915  vtxdun  29445  isoun  32658  cycpmfv3  33070  lbsdiflsp0  33598  sseqfv1  34356  reprsuc  34582  breprexplema  34597  cvmliftlem5  35261  fullfunfv  35920  finixpnum  37584  poimirlem1  37600  poimirlem2  37601  poimirlem3  37602  poimirlem4  37603  poimirlem6  37605  poimirlem7  37606  poimirlem11  37610  poimirlem12  37611  poimirlem16  37615  poimirlem17  37616  poimirlem19  37618  poimirlem22  37621  poimirlem23  37622  poimirlem28  37627  aacllem  49787