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Theorem fvun1 6746
 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 6442 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 1130 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 6442 . . . 4 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 1131 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 6444 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 6444 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 4177 . . . . . . 7 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2826 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 251 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 495 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1114 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 fvun 6745 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
132, 4, 11, 12syl21anc 836 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝑋) ∪ (𝐺𝑋)))
14 disjel 4390 . . . . . . . 8 (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → ¬ 𝑋𝐵)
1514adantl 485 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋𝐵)
166eleq2d 2901 . . . . . . . 8 (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺𝑋𝐵))
1716adantr 484 . . . . . . 7 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐺𝑋𝐵))
1815, 17mtbird 328 . . . . . 6 ((𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
19183adant1 1127 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ¬ 𝑋 ∈ dom 𝐺)
20 ndmfv 6692 . . . . 5 𝑋 ∈ dom 𝐺 → (𝐺𝑋) = ∅)
2119, 20syl 17 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝐺𝑋) = ∅)
2221uneq2d 4126 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = ((𝐹𝑋) ∪ ∅))
23 un0 4328 . . 3 ((𝐹𝑋) ∪ ∅) = (𝐹𝑋)
2422, 23syl6eq 2875 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝑋) ∪ (𝐺𝑋)) = (𝐹𝑋))
2513, 24eqtrd 2859 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ∪ cun 3918   ∩ cin 3919  ∅c0 4277  dom cdm 5543  Fun wfun 6338   Fn wfn 6339  ‘cfv 6344 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352 This theorem is referenced by:  fvun2  6747  enfixsn  8623  hashf1lem1  13821  ptunhmeo  22419  axlowdimlem6  26747  axlowdimlem8  26749  axlowdimlem11  26752  vtxdun  27277  isoun  30451  cycpmfv3  30792  lbsdiflsp0  31085  sseqfv1  31707  reprsuc  31946  breprexplema  31961  cvmliftlem5  32596  frrlem12  33194  noextenddif  33235  fullfunfv  33468  finixpnum  34988  poimirlem1  35004  poimirlem2  35005  poimirlem3  35006  poimirlem4  35007  poimirlem6  35009  poimirlem7  35010  poimirlem11  35014  poimirlem12  35015  poimirlem16  35019  poimirlem17  35020  poimirlem19  35022  poimirlem22  35025  poimirlem23  35026  poimirlem28  35031  fvun1d  39216  aacllem  45256
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