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Theorem fvun1 6411
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 6128 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1127 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 6128 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 1128 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 6130 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 6130 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3967 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2773 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 238 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 479 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1109 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 fvun 6410 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
132, 4, 11, 12syl21anc 1475 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
14 disjel 4167 . . . . . . . 8 (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → ¬ 𝑋𝐵)
1514adantl 467 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋𝐵)
166eleq2d 2836 . . . . . . . 8 (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺𝑋𝐵))
1716adantr 466 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐺𝑋𝐵))
1815, 17mtbird 314 . . . . . 6 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
19183adant1 1124 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20 ndmfv 6359 . . . . 5 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
2119, 20syl 17 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝐺𝑋) = ∅)
2221uneq2d 3918 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = ((𝐹𝑋) ∪ ∅))
23 un0 4111 . . 3 ((𝐹𝑋) ∪ ∅) = (𝐹𝑋)
2422, 23syl6eq 2821 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = (𝐹𝑋))
2513, 24eqtrd 2805 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cun 3721  cin 3722  c0 4063  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  fvun2  6412  enfixsn  8225  hashf1lem1  13441  xpsc0  16428  ptunhmeo  21832  axlowdimlem6  26048  axlowdimlem8  26050  axlowdimlem11  26053  vtxdun  26612  isoun  29819  sseqfv1  30791  reprsuc  31033  breprexplema  31048  cvmliftlem5  31609  noextenddif  32158  fullfunfv  32391  finixpnum  33727  poimirlem1  33743  poimirlem2  33744  poimirlem3  33745  poimirlem4  33746  poimirlem6  33748  poimirlem7  33749  poimirlem11  33753  poimirlem12  33754  poimirlem16  33758  poimirlem17  33759  poimirlem19  33761  poimirlem22  33764  poimirlem23  33765  poimirlem28  33770  aacllem  43078
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