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Theorem fvun1 5152
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 4910 . . 3 (𝐹 Fn A → Fun 𝐹)
213ad2ant1 907 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → Fun 𝐹)
3 fnfun 4910 . . 3 (𝐺 Fn B → Fun 𝐺)
433ad2ant2 908 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → Fun 𝐺)
5 fndm 4912 . . . . . . 7 (𝐹 Fn A → dom 𝐹 = A)
6 fndm 4912 . . . . . . 7 (𝐺 Fn B → dom 𝐺 = B)
75, 6ineqan12d 3108 . . . . . 6 ((𝐹 Fn A 𝐺 Fn B) → (dom 𝐹 ∩ dom 𝐺) = (AB))
87eqeq1d 2021 . . . . 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 1082 . 2 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 simp3r 915 . . 3 ((𝐹 Fn A 𝐺 Fn B ((AB) = ∅ 𝑋 A)) → 𝑋 A)
135eleq2d 2080 . . . 4 (𝐹 Fn A → (𝑋 dom 𝐹𝑋 A))
14133ad2ant1 907 . . 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 4858 . . . . . . 7 (((Fun 𝐹 Fun 𝐺) (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
17 ssun1 3074 . . . . . . . . 9 𝐹 ⊆ (𝐹𝐺)
18 dmss 4449 . . . . . . . . 9 (𝐹 ⊆ (𝐹𝐺) → dom 𝐹 ⊆ dom (𝐹𝐺))
1917, 18ax-mp 7 . . . . . . . 8 dom 𝐹 ⊆ dom (𝐹𝐺)
2019sseli 2909 . . . . . . 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 1080 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (Fun (𝐹𝐺) 𝑋 dom (𝐹𝐺)))
24 funfvdm 5149 . . . 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 4652 . . . . . 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 3554 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29 disjel 3242 . . . . . . . . 9 (((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹) → ¬ 𝑋 dom 𝐺)
30 ndmima 4617 . . . . . . . . 9 𝑋 dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
3129, 30syl 14 . . . . . . . 8 (((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32313ad2ant3 909 . . . . . . 7 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
3332uneq2d 3065 . . . . . 6 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34 un0 3219 . . . . . 6 ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
3533, 34syl6eq 2061 . . . . 5 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3635unieqd 3554 . . . 4 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3728, 36eqtrd 2045 . . 3 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {𝑋}))
38 funfvdm 5149 . . . . . 6 ((Fun 𝐹 𝑋 dom 𝐹) → (𝐹𝑋) = (𝐹 “ {𝑋}))
3938eqcomd 2018 . . . . 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 905 . . 3 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4225, 37, 413eqtrd 2049 . 2 ((Fun 𝐹 Fun 𝐺 ((dom 𝐹 ∩ dom 𝐺) = ∅ 𝑋 dom 𝐹)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
432, 4, 11, 15, 42syl112anc 1120 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 867   = wceq 1223   wcel 1366  cun 2883  cin 2884  wss 2885  c0 3192  {csn 3339   cuni 3543  dom cdm 4260  cima 4263  Fun wfun 4811   Fn wfn 4812  cfv 4817
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 529  ax-in2 530  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-fal 1229  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-dif 2888  df-un 2890  df-in 2892  df-ss 2899  df-nul 3193  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-fv 4825
This theorem is referenced by:  fvun2  5153
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