Theorem fv3 6663

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 6643	. . 3 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
2		biimpr 223	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑦 = 𝑧 → 𝐴𝐹𝑦))
3	2	alimi 1813	. . . . . . . . 9 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦))
4		breq2 5034	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑧))
5	4	equsalvw 2010	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
6	3, 5	sylib 221	. . . . . . . 8 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → 𝐴𝐹𝑧)
7	6	anim2i 619	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
8	7	eximi 1836	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
9		elequ2 2126	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
10		breq2 5034	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
11	9, 10	anbi12d 633	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)))
12	11	cbvexvw 2044	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
13	8, 12	sylib 221	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
14		exsimpr 1870	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
15		eu6 2634	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
16	14, 15	sylibr 237	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
17	13, 16	jca 515	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
18		nfeu1 2649	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
19		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑧
20		nfa1 2152	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)
21	19, 20	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
22	21	nfex 2332	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
23	18, 22	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
24		biimp 218	. . . . . . . . . . . . 13 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → 𝑦 = 𝑧))
25		ax9 2125	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
26	24, 25	syl6 35	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)))
27	26	impcomd 415	. . . . . . . . . . 11 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
28	27	sps 2182	. . . . . . . . . 10 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
29	28	anc2ri 560	. . . . . . . . 9 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
30	29	com12 32	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
31	30	eximdv 1918	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
32	15, 31	syl5bi 245	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
33	23, 32	exlimi 2215	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
34	33	imp 410	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
35	17, 34	impbii 212	. . 3 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
36	1, 35	bitri 278	. 2 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
37	36	abbi2i 2929	1 ⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  {cab 2776   class class class wbr 5030  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by: (None)