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Theorem fvun1 5379
 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
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 fnfun 5181 . . . . 5 (F Fn A → Fun F)
213ad2ant1 976 . . . 4 ((F Fn A G Fn B ((AB) = X A)) → Fun F)
3 fnfun 5181 . . . . 5 (G Fn B → Fun G)
433ad2ant2 977 . . . 4 ((F Fn A G Fn B ((AB) = X A)) → Fun G)
5 fndm 5182 . . . . . . . . 9 (F Fn A → dom F = A)
6 fndm 5182 . . . . . . . . 9 (G Fn B → dom G = B)
7 ineq12 3452 . . . . . . . . 9 ((dom F = A dom G = B) → (dom F ∩ dom G) = (AB))
85, 6, 7syl2an 463 . . . . . . . 8 ((F Fn A G Fn B) → (dom F ∩ dom G) = (AB))
98eqeq1d 2361 . . . . . . 7 ((F Fn A G Fn B) → ((dom F ∩ dom G) = ↔ (AB) = ))
109biimprd 214 . . . . . 6 ((F Fn A G Fn B) → ((AB) = → (dom F ∩ dom G) = ))
1110adantrd 454 . . . . 5 ((F Fn A G Fn B) → (((AB) = X A) → (dom F ∩ dom G) = ))
12113impia 1148 . . . 4 ((F Fn A G Fn B ((AB) = X A)) → (dom F ∩ dom G) = )
13 fvun 5378 . . . 4 (((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) ‘X) = ((FX) ∪ (GX)))
142, 4, 12, 13syl21anc 1181 . . 3 ((F Fn A G Fn B ((AB) = X A)) → ((FG) ‘X) = ((FX) ∪ (GX)))
15 disj 3591 . . . . . . . . 9 ((AB) = x A ¬ x B)
16 eleq1 2413 . . . . . . . . . . 11 (x = X → (x BX B))
1716notbid 285 . . . . . . . . . 10 (x = X → (¬ x B ↔ ¬ X B))
1817rspccv 2952 . . . . . . . . 9 (x A ¬ x B → (X A → ¬ X B))
1915, 18sylbi 187 . . . . . . . 8 ((AB) = → (X A → ¬ X B))
2019imp 418 . . . . . . 7 (((AB) = X A) → ¬ X B)
21203ad2ant3 978 . . . . . 6 ((F Fn A G Fn B ((AB) = X A)) → ¬ X B)
2263ad2ant2 977 . . . . . . 7 ((F Fn A G Fn B ((AB) = X A)) → dom G = B)
2322eleq2d 2420 . . . . . 6 ((F Fn A G Fn B ((AB) = X A)) → (X dom GX B))
2421, 23mtbird 292 . . . . 5 ((F Fn A G Fn B ((AB) = X A)) → ¬ X dom G)
25 ndmfv 5349 . . . . 5 X dom G → (GX) = )
2624, 25syl 15 . . . 4 ((F Fn A G Fn B ((AB) = X A)) → (GX) = )
2726uneq2d 3418 . . 3 ((F Fn A G Fn B ((AB) = X A)) → ((FX) ∪ (GX)) = ((FX) ∪ ))
2814, 27eqtrd 2385 . 2 ((F Fn A G Fn B ((AB) = X A)) → ((FG) ‘X) = ((FX) ∪ ))
29 un0 3575 . 2 ((FX) ∪ ) = (FX)
3028, 29syl6eq 2401 1 ((F Fn A G Fn B ((AB) = X A)) → ((FG) ‘X) = (FX))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795 This theorem is referenced by:  fvun2  5380  fvfullfun  5864
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