Theorem fv3 5342

Description: Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
fv3	⊢ (F ‘A) = {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)}

Distinct variable groups:   x,y,F   x,A,y

Proof of Theorem fv3

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfv 5327	. . 3 ⊢ (x ∈ (F ‘A) ↔ ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)))
2		bi2 189	. . . . . . . . . 10 ⊢ ((AFy ↔ y = z) → (y = z → AFy))
3	2	alimi 1559	. . . . . . . . 9 ⊢ (∀y(AFy ↔ y = z) → ∀y(y = z → AFy))
4		vex 2863	. . . . . . . . . 10 ⊢ z ∈ V
5		breq2 4644	. . . . . . . . . 10 ⊢ (y = z → (AFy ↔ AFz))
6	4, 5	ceqsalv 2886	. . . . . . . . 9 ⊢ (∀y(y = z → AFy) ↔ AFz)
7	3, 6	sylib 188	. . . . . . . 8 ⊢ (∀y(AFy ↔ y = z) → AFz)
8	7	anim2i 552	. . . . . . 7 ⊢ ((x ∈ z ∧ ∀y(AFy ↔ y = z)) → (x ∈ z ∧ AFz))
9	8	eximi 1576	. . . . . 6 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → ∃z(x ∈ z ∧ AFz))
10		elequ2 1715	. . . . . . . 8 ⊢ (z = y → (x ∈ z ↔ x ∈ y))
11		breq2 4644	. . . . . . . 8 ⊢ (z = y → (AFz ↔ AFy))
12	10, 11	anbi12d 691	. . . . . . 7 ⊢ (z = y → ((x ∈ z ∧ AFz) ↔ (x ∈ y ∧ AFy)))
13	12	cbvexv 2003	. . . . . 6 ⊢ (∃z(x ∈ z ∧ AFz) ↔ ∃y(x ∈ y ∧ AFy))
14	9, 13	sylib 188	. . . . 5 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → ∃y(x ∈ y ∧ AFy))
15		19.40 1609	. . . . . . 7 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → (∃z x ∈ z ∧ ∃z∀y(AFy ↔ y = z)))
16	15	simprd 449	. . . . . 6 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → ∃z∀y(AFy ↔ y = z))
17		df-eu 2208	. . . . . 6 ⊢ (∃!y AFy ↔ ∃z∀y(AFy ↔ y = z))
18	16, 17	sylibr 203	. . . . 5 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → ∃!y AFy)
19	14, 18	jca 518	. . . 4 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) → (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy))
20		nfeu1 2214	. . . . . . 7 ⊢ Ⅎy∃!y AFy
21		nfv 1619	. . . . . . . . 9 ⊢ Ⅎy x ∈ z
22		nfa1 1788	. . . . . . . . 9 ⊢ Ⅎy∀y(AFy ↔ y = z)
23	21, 22	nfan 1824	. . . . . . . 8 ⊢ Ⅎy(x ∈ z ∧ ∀y(AFy ↔ y = z))
24	23	nfex 1843	. . . . . . 7 ⊢ Ⅎy∃z(x ∈ z ∧ ∀y(AFy ↔ y = z))
25	20, 24	nfim 1813	. . . . . 6 ⊢ Ⅎy(∃!y AFy → ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)))
26		bi1 178	. . . . . . . . . . . . . 14 ⊢ ((AFy ↔ y = z) → (AFy → y = z))
27		ax-14 1714	. . . . . . . . . . . . . 14 ⊢ (y = z → (x ∈ y → x ∈ z))
28	26, 27	syl6 29	. . . . . . . . . . . . 13 ⊢ ((AFy ↔ y = z) → (AFy → (x ∈ y → x ∈ z)))
29	28	com23 72	. . . . . . . . . . . 12 ⊢ ((AFy ↔ y = z) → (x ∈ y → (AFy → x ∈ z)))
30	29	imp3a 420	. . . . . . . . . . 11 ⊢ ((AFy ↔ y = z) → ((x ∈ y ∧ AFy) → x ∈ z))
31	30	sps 1754	. . . . . . . . . 10 ⊢ (∀y(AFy ↔ y = z) → ((x ∈ y ∧ AFy) → x ∈ z))
32	31	anc2ri 541	. . . . . . . . 9 ⊢ (∀y(AFy ↔ y = z) → ((x ∈ y ∧ AFy) → (x ∈ z ∧ ∀y(AFy ↔ y = z))))
33	32	com12 27	. . . . . . . 8 ⊢ ((x ∈ y ∧ AFy) → (∀y(AFy ↔ y = z) → (x ∈ z ∧ ∀y(AFy ↔ y = z))))
34	33	eximdv 1622	. . . . . . 7 ⊢ ((x ∈ y ∧ AFy) → (∃z∀y(AFy ↔ y = z) → ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z))))
35	17, 34	syl5bi 208	. . . . . 6 ⊢ ((x ∈ y ∧ AFy) → (∃!y AFy → ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z))))
36	25, 35	exlimi 1803	. . . . 5 ⊢ (∃y(x ∈ y ∧ AFy) → (∃!y AFy → ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z))))
37	36	imp 418	. . . 4 ⊢ ((∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy) → ∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)))
38	19, 37	impbii 180	. . 3 ⊢ (∃z(x ∈ z ∧ ∀y(AFy ↔ y = z)) ↔ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy))
39	1, 38	bitri 240	. 2 ⊢ (x ∈ (F ‘A) ↔ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy))
40	39	abbi2i 2465	1 ⊢ (F ‘A) = {x ∣ (∃y(x ∈ y ∧ AFy) ∧ ∃!y AFy)}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339   class class class wbr 4640   ‘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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796

This theorem is referenced by:  tz6.12-2  5347