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Theorem fvun1 6160
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 5884 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1074 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 5884 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 1075 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 5886 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 5886 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3773 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2607 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 236 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 482 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1252 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 fvun 6159 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
132, 4, 11, 12syl21anc 1316 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
14 disjel 3970 . . . . . . . 8 (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → ¬ 𝑋𝐵)
1514adantl 480 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋𝐵)
166eleq2d 2668 . . . . . . . 8 (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺𝑋𝐵))
1716adantr 479 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐺𝑋𝐵))
1815, 17mtbird 313 . . . . . 6 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
19183adant1 1071 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20 ndmfv 6109 . . . . 5 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
2119, 20syl 17 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝐺𝑋) = ∅)
2221uneq2d 3724 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = ((𝐹𝑋) ∪ ∅))
23 un0 3914 . . 3 ((𝐹𝑋) ∪ ∅) = (𝐹𝑋)
2422, 23syl6eq 2655 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = (𝐹𝑋))
2513, 24eqtrd 2639 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  cun 3533  cin 3534  c0 3869  dom cdm 5024  Fun wfun 5780   Fn wfn 5781  cfv 5786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-fv 5794
This theorem is referenced by:  fvun2  6161  enfixsn  7927  hashf1lem1  13044  xpsc0  15985  ptunhmeo  21359  axlowdimlem6  25541  axlowdimlem8  25543  axlowdimlem11  25546  constr3lem4  25937  vdgrun  26190  vdgrfiun  26191  isoun  28664  sseqfv1  29580  cvmliftlem5  30327  fullfunfv  31026  finixpnum  32363  poimirlem1  32379  poimirlem2  32380  poimirlem3  32381  poimirlem4  32382  poimirlem6  32384  poimirlem7  32385  poimirlem11  32389  poimirlem12  32390  poimirlem16  32394  poimirlem17  32395  poimirlem19  32397  poimirlem22  32400  poimirlem23  32401  poimirlem28  32406  vtxdun  40694  aacllem  42315
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