Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvun1 Structured version   Visualization version   GIF version

Theorem fvun1 6179
 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 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))

Proof of Theorem fvun1
StepHypRef Expression
1 fnfun 5902 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1075 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 5902 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 1076 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 5904 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 5904 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3778 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2612 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 237 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 483 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1253 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 fvun 6178 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
132, 4, 11, 12syl21anc 1317 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
14 disjel 3975 . . . . . . . 8 (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → ¬ 𝑋𝐵)
1514adantl 481 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋𝐵)
166eleq2d 2673 . . . . . . . 8 (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺𝑋𝐵))
1716adantr 480 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐺𝑋𝐵))
1815, 17mtbird 314 . . . . . 6 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
19183adant1 1072 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20 ndmfv 6128 . . . . 5 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
2119, 20syl 17 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝐺𝑋) = ∅)
2221uneq2d 3729 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = ((𝐹𝑋) ∪ ∅))
23 un0 3919 . . 3 ((𝐹𝑋) ∪ ∅) = (𝐹𝑋)
2422, 23syl6eq 2660 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = (𝐹𝑋))
2513, 24eqtrd 2644 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  fvun2  6180  enfixsn  7954  hashf1lem1  13096  xpsc0  16043  ptunhmeo  21421  axlowdimlem6  25627  axlowdimlem8  25629  axlowdimlem11  25632  constr3lem4  26175  vdgrun  26428  vdgrfiun  26429  isoun  28862  sseqfv1  29778  cvmliftlem5  30525  fullfunfv  31224  finixpnum  32564  poimirlem1  32580  poimirlem2  32581  poimirlem3  32582  poimirlem4  32583  poimirlem6  32585  poimirlem7  32586  poimirlem11  32590  poimirlem12  32591  poimirlem16  32595  poimirlem17  32596  poimirlem19  32598  poimirlem22  32601  poimirlem23  32602  poimirlem28  32607  vtxdun  40696  aacllem  42356
 Copyright terms: Public domain W3C validator