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Theorem fvun1 5182
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 A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))

Proof of Theorem fvun1
StepHypRef Expression
1 fnfun 4939 . . 3 (𝐹 Fn A → Fun 𝐹)
213ad2ant1 924 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → Fun 𝐹)
3 fnfun 4939 . . 3 (𝐺 Fn B → Fun 𝐺)
433ad2ant2 925 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → Fun 𝐺)
5 fndm 4941 . . . . . . 7 (𝐹 Fn A → dom 𝐹 = A)
6 fndm 4941 . . . . . . 7 (𝐺 Fn B → dom 𝐺 = B)
75, 6ineqan12d 3134 . . . . . 6 ((𝐹 Fn A 𝐺 Fn B) → (dom 𝐹 ∩ dom 𝐺) = (AB))
87eqeq1d 2045 . . . . 5 ((𝐹 Fn A 𝐺 Fn B) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (AB) = ∅))
98biimprd 147 . . . 4 ((𝐹 Fn A 𝐺 Fn B) → ((AB) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 264 . . 3 ((𝐹 Fn A 𝐺 Fn B) → (((AB) = ∅ 𝑋 A) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1100 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 simp3r 932 . . 3 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → 𝑋 A)
135eleq2d 2104 . . . 4 (𝐹 Fn A → (𝑋 dom 𝐹𝑋 A))
14133ad2ant1 924 . . 3 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → (𝑋 dom 𝐹𝑋 A))
1512, 14mpbird 156 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → 𝑋 dom 𝐹)
16 funun 4887 . . . . . . 7 (((Fun 𝐹 Fun 𝐺) (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
17 ssun1 3100 . . . . . . . . 9 𝐹 ⊆ (𝐹𝐺)
18 dmss 4477 . . . . . . . . 9 (𝐹 ⊆ (𝐹𝐺) → dom 𝐹 ⊆ dom (𝐹𝐺))
1917, 18ax-mp 7 . . . . . . . 8 dom 𝐹 ⊆ dom (𝐹𝐺)
2019sseli 2935 . . . . . . 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 1098 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (Fun (𝐹𝐺) 𝑋 dom (𝐹𝐺)))
24 funfvdm 5179 . . . 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 4680 . . . . . 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 3582 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29 disjel 3268 . . . . . . . . 9 (((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹) → ¬ 𝑋 dom 𝐺)
30 ndmima 4645 . . . . . . . . 9 𝑋 dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
3129, 30syl 14 . . . . . . . 8 (((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32313ad2ant3 926 . . . . . . 7 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
3332uneq2d 3091 . . . . . 6 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34 un0 3245 . . . . . 6 ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
3533, 34syl6eq 2085 . . . . 5 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3635unieqd 3582 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3728, 36eqtrd 2069 . . 3 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {𝑋}))
38 funfvdm 5179 . . . . . 6 ((Fun 𝐹 𝑋 dom 𝐹) → (𝐹𝑋) = (𝐹 “ {𝑋}))
3938eqcomd 2042 . . . . 5 ((Fun 𝐹 𝑋 dom 𝐹) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4039adantrl 447 . . . 4 ((Fun 𝐹 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
41403adant2 922 . . 3 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4225, 37, 413eqtrd 2073 . 2 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
432, 4, 11, 15, 42syl112anc 1138 1 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cun 2909  cin 2910  wss 2911  c0 3218  {csn 3367   cuni 3571  dom cdm 4288  cima 4291  Fun wfun 4839   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fvun2  5183
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