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Theorem fvun1 5239
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 4996 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 925 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 4996 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 926 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 4998 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 4998 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3140 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2048 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 147 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 264 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1101 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 simp3r 933 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋𝐴)
135eleq2d 2107 . . . 4 (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹𝑋𝐴))
14133ad2ant1 925 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐹𝑋𝐴))
1512, 14mpbird 156 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋 ∈ dom 𝐹)
16 funun 4944 . . . . . . 7 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
17 ssun1 3106 . . . . . . . . 9 𝐹 ⊆ (𝐹𝐺)
18 dmss 4534 . . . . . . . . 9 (𝐹 ⊆ (𝐹𝐺) → dom 𝐹 ⊆ dom (𝐹𝐺))
1917, 18ax-mp 7 . . . . . . . 8 dom 𝐹 ⊆ dom (𝐹𝐺)
2019sseli 2941 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺))
2116, 20anim12i 321 . . . . . 6 ((((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑋 ∈ dom 𝐹) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
2221anasss 379 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺) ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
23223impa 1099 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
24 funfvdm 5236 . . . 4 ((Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝐺) “ {𝑋}))
2523, 24syl 14 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺)‘𝑋) = ((𝐹𝐺) “ {𝑋}))
26 imaundir 4737 . . . . . 6 ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋}))
2726a1i 9 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
2827unieqd 3591 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29 disjel 3274 . . . . . . . . 9 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30 ndmima 4702 . . . . . . . . 9 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
3129, 30syl 14 . . . . . . . 8 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32313ad2ant3 927 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
3332uneq2d 3097 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34 un0 3251 . . . . . 6 ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
3533, 34syl6eq 2088 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3635unieqd 3591 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3728, 36eqtrd 2072 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {𝑋}))
38 funfvdm 5236 . . . . . 6 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝐹 “ {𝑋}))
3938eqcomd 2045 . . . . 5 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4039adantrl 447 . . . 4 ((Fun 𝐹 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
41403adant2 923 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4225, 37, 413eqtrd 2076 . 2 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
432, 4, 11, 15, 42syl112anc 1139 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  cun 2915  cin 2916  wss 2917  c0 3224  {csn 3375   cuni 3580  dom cdm 4345  cima 4348  Fun wfun 4896   Fn wfn 4897  cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  fvun2  5240
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