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Theorem fvun1 5453
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 5188 . . 3 (𝐹 Fn 𝐴 → Fun 𝐹)
213ad2ant1 985 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐹)
3 fnfun 5188 . . 3 (𝐺 Fn 𝐵 → Fun 𝐺)
433ad2ant2 986 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → Fun 𝐺)
5 fndm 5190 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6 fndm 5190 . . . . . . 7 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
75, 6ineqan12d 3247 . . . . . 6 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
87eqeq1d 2124 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴𝐵) = ∅))
98biimprd 157 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → ((𝐴𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅))
109adantrd 275 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (((𝐴𝐵) = ∅ ∧ 𝑋𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅))
11103impia 1161 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅)
12 simp3r 993 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋𝐴)
135eleq2d 2185 . . . 4 (𝐹 Fn 𝐴 → (𝑋 ∈ dom 𝐹𝑋𝐴))
14133ad2ant1 985 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → (𝑋 ∈ dom 𝐹𝑋𝐴))
1512, 14mpbird 166 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → 𝑋 ∈ dom 𝐹)
16 funun 5135 . . . . . . 7 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹𝐺))
17 ssun1 3207 . . . . . . . . 9 𝐹 ⊆ (𝐹𝐺)
18 dmss 4706 . . . . . . . . 9 (𝐹 ⊆ (𝐹𝐺) → dom 𝐹 ⊆ dom (𝐹𝐺))
1917, 18ax-mp 5 . . . . . . . 8 dom 𝐹 ⊆ dom (𝐹𝐺)
2019sseli 3061 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺))
2116, 20anim12i 334 . . . . . 6 ((((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑋 ∈ dom 𝐹) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
2221anasss 394 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺) ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
23223impa 1159 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (Fun (𝐹𝐺) ∧ 𝑋 ∈ dom (𝐹𝐺)))
24 funfvdm 5450 . . . 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 4920 . . . . . 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 3715 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})))
29 disjel 3385 . . . . . . . . 9 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → ¬ 𝑋 ∈ dom 𝐺)
30 ndmima 4884 . . . . . . . . 9 𝑋 ∈ dom 𝐺 → (𝐺 “ {𝑋}) = ∅)
3129, 30syl 14 . . . . . . . 8 (((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹) → (𝐺 “ {𝑋}) = ∅)
32313ad2ant3 987 . . . . . . 7 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐺 “ {𝑋}) = ∅)
3332uneq2d 3198 . . . . . 6 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = ((𝐹 “ {𝑋}) ∪ ∅))
34 un0 3364 . . . . . 6 ((𝐹 “ {𝑋}) ∪ ∅) = (𝐹 “ {𝑋})
3533, 34syl6eq 2164 . . . . 5 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3635unieqd 3715 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹 “ {𝑋}) ∪ (𝐺 “ {𝑋})) = (𝐹 “ {𝑋}))
3728, 36eqtrd 2148 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺) “ {𝑋}) = (𝐹 “ {𝑋}))
38 funfvdm 5450 . . . . . 6 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = (𝐹 “ {𝑋}))
3938eqcomd 2121 . . . . 5 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4039adantrl 467 . . . 4 ((Fun 𝐹 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
41403adant2 983 . . 3 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → (𝐹 “ {𝑋}) = (𝐹𝑋))
4225, 37, 413eqtrd 2152 . 2 ((Fun 𝐹 ∧ Fun 𝐺 ∧ ((dom 𝐹 ∩ dom 𝐺) = ∅ ∧ 𝑋 ∈ dom 𝐹)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
432, 4, 11, 15, 42syl112anc 1203 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 945   = wceq 1314  wcel 1463  cun 3037  cin 3038  wss 3039  c0 3331  {csn 3495   cuni 3704  dom cdm 4507  cima 4510  Fun wfun 5085   Fn wfn 5086  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  fvun2  5454  caseinl  6942
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