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Theorem fvun1 5266
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 5023 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 936 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 5023 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 937 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 5025 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 5025 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3167 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2064 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 151 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 268 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1112 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 simp3r 944 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋𝐴)
135eleq2d 2123 . . . 4 (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹𝑋𝐴))
14133ad2ant1 936 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐹𝑋𝐴))
1512, 14mpbird 160 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋 ∈ dom 𝐹)
16 funun 4971 . . . . . . 7 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
17 ssun1 3133 . . . . . . . . 9 𝐹 ⊆ (𝐹𝐺)
18 dmss 4561 . . . . . . . . 9 (𝐹 ⊆ (𝐹𝐺) → dom 𝐹 ⊆ dom (𝐹𝐺))
1917, 18ax-mp 7 . . . . . . . 8 dom 𝐹 ⊆ dom (𝐹𝐺)
2019sseli 2968 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺))
2116, 20anim12i 325 . . . . . 6 ((((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑋 ∈ dom 𝐹) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
2221anasss 385 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺) ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
23223impa 1110 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
24 funfvdm 5263 . . . 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 4764 . . . . . 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 3618 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29 disjel 3301 . . . . . . . . 9 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30 ndmima 4729 . . . . . . . . 9 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
3129, 30syl 14 . . . . . . . 8 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32313ad2ant3 938 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
3332uneq2d 3124 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34 un0 3278 . . . . . 6 ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
3533, 34syl6eq 2104 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3635unieqd 3618 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3728, 36eqtrd 2088 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {𝑋}))
38 funfvdm 5263 . . . . . 6 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝐹 “ {𝑋}))
3938eqcomd 2061 . . . . 5 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4039adantrl 455 . . . 4 ((Fun 𝐹 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
41403adant2 934 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4225, 37, 413eqtrd 2092 . 2 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
432, 4, 11, 15, 42syl112anc 1150 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  cun 2942  cin 2943  wss 2944  c0 3251  {csn 3402   cuni 3607  dom cdm 4372  cima 4375  Fun wfun 4923   Fn wfn 4924  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by:  fvun2  5267
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