Theorem funiunfv 5468

Description: The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to F Fn A, the theorem can be proved without this dependency. (Contributed by set.mm contributors, 26-Mar-2006.)

Assertion

Ref	Expression
funiunfv	⊢ (Fun F → ∪x ∈ A (F ‘x) = ∪(F “ A))

Distinct variable groups:   x,A   x,F

Proof of Theorem funiunfv

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5329	. . . . 5 ⊢ (y = x → (F ‘y) = (F ‘x))
2		eqid 2353	. . . . 5 ⊢ {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} = {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))}
3		fvex 5340	. . . . 5 ⊢ (F ‘x) ∈ V
4	1, 2, 3	fvopab4 5390	. . . 4 ⊢ (x ∈ A → ({⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} ‘x) = (F ‘x))
5	4	iuneq2i 3988	. . 3 ⊢ ∪x ∈ A ({⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} ‘x) = ∪x ∈ A (F ‘x)
6		fvex 5340	. . . . 5 ⊢ (F ‘y) ∈ V
7	6, 2	fnopab2 5209	. . . 4 ⊢ {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} Fn A
8		fniunfv 5467	. . . 4 ⊢ ({⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} Fn A → ∪x ∈ A ({⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} ‘x) = ∪ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))})
9	7, 8	ax-mp 5	. . 3 ⊢ ∪x ∈ A ({⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} ‘x) = ∪ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))}
10	5, 9	eqtr3i 2375	. 2 ⊢ ∪x ∈ A (F ‘x) = ∪ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))}
11		rnopab2 4969	. . . 4 ⊢ ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} = {z ∣ ∃y ∈ A z = (F ‘y)}
12	11	unieqi 3902	. . 3 ⊢ ∪ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} = ∪{z ∣ ∃y ∈ A z = (F ‘y)}
13		eqcom 2355	. . . . . . . . 9 ⊢ (z = (F ‘y) ↔ (F ‘y) = z)
14		idd 21	. . . . . . . . . 10 ⊢ ((Fun F ∧ w ∈ z) → ((F ‘y) = z → (F ‘y) = z))
15		funbrfv 5357	. . . . . . . . . . 11 ⊢ (Fun F → (yFz → (F ‘y) = z))
16	15	adantr 451	. . . . . . . . . 10 ⊢ ((Fun F ∧ w ∈ z) → (yFz → (F ‘y) = z))
17		n0i 3556	. . . . . . . . . . . . 13 ⊢ (w ∈ z → ¬ z = ∅)
18		ndmfv 5350	. . . . . . . . . . . . . . 15 ⊢ (¬ y ∈ dom F → (F ‘y) = ∅)
19		eqeq1 2359	. . . . . . . . . . . . . . 15 ⊢ ((F ‘y) = z → ((F ‘y) = ∅ ↔ z = ∅))
20	18, 19	syl5ib 210	. . . . . . . . . . . . . 14 ⊢ ((F ‘y) = z → (¬ y ∈ dom F → z = ∅))
21	20	con1d 116	. . . . . . . . . . . . 13 ⊢ ((F ‘y) = z → (¬ z = ∅ → y ∈ dom F))
22	17, 21	mpan9 455	. . . . . . . . . . . 12 ⊢ ((w ∈ z ∧ (F ‘y) = z) → y ∈ dom F)
23		funbrfvb 5361	. . . . . . . . . . . 12 ⊢ ((Fun F ∧ y ∈ dom F) → ((F ‘y) = z ↔ yFz))
24	22, 23	sylan2 460	. . . . . . . . . . 11 ⊢ ((Fun F ∧ (w ∈ z ∧ (F ‘y) = z)) → ((F ‘y) = z ↔ yFz))
25	24	expr 598	. . . . . . . . . 10 ⊢ ((Fun F ∧ w ∈ z) → ((F ‘y) = z → ((F ‘y) = z ↔ yFz)))
26	14, 16, 25	pm5.21ndd 343	. . . . . . . . 9 ⊢ ((Fun F ∧ w ∈ z) → ((F ‘y) = z ↔ yFz))
27	13, 26	syl5bb 248	. . . . . . . 8 ⊢ ((Fun F ∧ w ∈ z) → (z = (F ‘y) ↔ yFz))
28	27	rexbidv 2636	. . . . . . 7 ⊢ ((Fun F ∧ w ∈ z) → (∃y ∈ A z = (F ‘y) ↔ ∃y ∈ A yFz))
29	28	pm5.32da 622	. . . . . 6 ⊢ (Fun F → ((w ∈ z ∧ ∃y ∈ A z = (F ‘y)) ↔ (w ∈ z ∧ ∃y ∈ A yFz)))
30	29	exbidv 1626	. . . . 5 ⊢ (Fun F → (∃z(w ∈ z ∧ ∃y ∈ A z = (F ‘y)) ↔ ∃z(w ∈ z ∧ ∃y ∈ A yFz)))
31		eluniab 3904	. . . . 5 ⊢ (w ∈ ∪{z ∣ ∃y ∈ A z = (F ‘y)} ↔ ∃z(w ∈ z ∧ ∃y ∈ A z = (F ‘y)))
32		eluni 3895	. . . . . 6 ⊢ (w ∈ ∪(F “ A) ↔ ∃z(w ∈ z ∧ z ∈ (F “ A)))
33		elima 4755	. . . . . . . 8 ⊢ (z ∈ (F “ A) ↔ ∃y ∈ A yFz)
34	33	anbi2i 675	. . . . . . 7 ⊢ ((w ∈ z ∧ z ∈ (F “ A)) ↔ (w ∈ z ∧ ∃y ∈ A yFz))
35	34	exbii 1582	. . . . . 6 ⊢ (∃z(w ∈ z ∧ z ∈ (F “ A)) ↔ ∃z(w ∈ z ∧ ∃y ∈ A yFz))
36	32, 35	bitri 240	. . . . 5 ⊢ (w ∈ ∪(F “ A) ↔ ∃z(w ∈ z ∧ ∃y ∈ A yFz))
37	30, 31, 36	3bitr4g 279	. . . 4 ⊢ (Fun F → (w ∈ ∪{z ∣ ∃y ∈ A z = (F ‘y)} ↔ w ∈ ∪(F “ A)))
38	37	eqrdv 2351	. . 3 ⊢ (Fun F → ∪{z ∣ ∃y ∈ A z = (F ‘y)} = ∪(F “ A))
39	12, 38	syl5eq 2397	. 2 ⊢ (Fun F → ∪ran {⟨y, z⟩ ∣ (y ∈ A ∧ z = (F ‘y))} = ∪(F “ A))
40	10, 39	syl5eq 2397	1 ⊢ (Fun F → ∪x ∈ A (F ‘x) = ∪(F “ A))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  ∅c0 3551  ∪cuni 3892  ∪ciun 3970  {copab 4623   class class class wbr 4640   “ cima 4723  dom cdm 4773  ran crn 4774  Fun wfun 4776   Fn wfn 4777   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  funiunfvf  5469  eluniima  5470