Theorem fv3 5484

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 5459	. . 3 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
2		biimpr 129	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑦 = 𝑧 → 𝐴𝐹𝑦))
3	2	alimi 1432	. . . . . . . . 9 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦))
4		vex 2712	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
5		breq2 3965	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑧))
6	4, 5	ceqsalv 2739	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
7	3, 6	sylib 121	. . . . . . . 8 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → 𝐴𝐹𝑧)
8	7	anim2i 340	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
9	8	eximi 1577	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
10		elequ2 2130	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
11		breq2 3965	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
12	10, 11	anbi12d 465	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)))
13	12	cbvexv 1895	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
14	9, 13	sylib 121	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
15		exsimpr 1595	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
16		df-eu 2006	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
17	15, 16	sylibr 133	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
18	14, 17	jca 304	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
19		nfeu1 2014	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
20		nfv 1505	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑧
21		nfa1 1518	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)
22	20, 21	nfan 1542	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
23	22	nfex 1614	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
24	19, 23	nfim 1549	. . . . . 6 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
25		biimp 117	. . . . . . . . . . . . . 14 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → 𝑦 = 𝑧))
26		ax-14 2128	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
27	25, 26	syl6 33	. . . . . . . . . . . . 13 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)))
28	27	com23 78	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑦 → (𝐴𝐹𝑦 → 𝑥 ∈ 𝑧)))
29	28	impd 252	. . . . . . . . . . 11 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
30	29	sps 1514	. . . . . . . . . 10 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
31	30	anc2ri 328	. . . . . . . . 9 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
32	31	com12 30	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
33	32	eximdv 1857	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
34	16, 33	syl5bi 151	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
35	24, 34	exlimi 1571	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
36	35	imp 123	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
37	18, 36	impbii 125	. . 3 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
38	1, 37	bitri 183	. 2 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
39	38	abbi2i 2269	1 ⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469  ∃!weu 2003   ∈ wcel 2125  {cab 2140   class class class wbr 3961  ‘cfv 5163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rex 2438  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-iota 5128  df-fv 5171

This theorem is referenced by: (None)