Theorem fvun 6756

Description: Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)

Assertion

Ref	Expression
fvun	⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Proof of Theorem fvun

Step	Hyp	Ref	Expression
1		funun 6403	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
2		funfv 6753	. . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → ((𝐹 ∪ 𝐺)‘𝐴) = ∪ ((𝐹 ∪ 𝐺) “ {𝐴}))
3	1, 2	syl 17	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ∪ ((𝐹 ∪ 𝐺) “ {𝐴}))
4		imaundir 6012	. . . 4 ⊢ ((𝐹 ∪ 𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
5	4	a1i 11	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
6	5	unieqd 4855	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 ∪ 𝐺) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7		uniun 4864	. . 3 ⊢ ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴}))
8		funfv 6753	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
9	8	eqcomd 2830	. . . . . 6 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴))
10		funfv 6753	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺‘𝐴) = ∪ (𝐺 “ {𝐴}))
11	10	eqcomd 2830	. . . . . 6 ⊢ (Fun 𝐺 → ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴))
12	9, 11	anim12i 614	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)))
13	12	adantr 483	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)))
14		uneq12 4137	. . . 4 ⊢ ((∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)) → (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
15	13, 14	syl 17	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
16	7, 15	syl5eq 2871	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
17	3, 6, 16	3eqtrd 2863	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∪ cun 3937   ∩ cin 3938  ∅c0 4294  {csn 4570  ∪ cuni 4841  dom cdm 5558   “ cima 5561  Fun wfun 6352  ‘cfv 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366

This theorem is referenced by:  fvun1  6757  undifixp  8501