Theorem fv3 5609

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 5584	. . 3 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
2		biimpr 130	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝑦 = 𝑧 → 𝐴𝐹𝑦))
3	2	alimi 1479	. . . . . . . . 9 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦))
4		vex 2776	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
5		breq2 4052	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑧))
6	4, 5	ceqsalv 2804	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑧 → 𝐴𝐹𝑦) ↔ 𝐴𝐹𝑧)
7	3, 6	sylib 122	. . . . . . . 8 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → 𝐴𝐹𝑧)
8	7	anim2i 342	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
9	8	eximi 1624	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧))
10		elequ2 2182	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
11		breq2 4052	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
12	10, 11	anbi12d 473	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦)))
13	12	cbvexv 1943	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ 𝐴𝐹𝑧) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
14	9, 13	sylib 122	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦))
15		exsimpr 1642	. . . . . 6 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
16		df-eu 2058	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
17	15, 16	sylibr 134	. . . . 5 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → ∃!𝑦 𝐴𝐹𝑦)
18	14, 17	jca 306	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
19		nfeu1 2066	. . . . . . 7 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
20		nfv 1552	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑧
21		nfa1 1565	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)
22	20, 21	nfan 1589	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
23	22	nfex 1661	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))
24	19, 23	nfim 1596	. . . . . 6 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
25		biimp 118	. . . . . . . . . . . . . 14 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → (𝐴𝐹𝑦 → 𝑦 = 𝑧))
26		ax-14 2180	. . . . . . . . . . . . . 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 254	. . . . . . . . . . 11 ⊢ ((𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
30	29	sps 1561	. . . . . . . . . 10 ⊢ (∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → 𝑥 ∈ 𝑧))
31	30	anc2ri 330	. . . . . . . . 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 1904	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃𝑧∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
34	16, 33	biimtrid 152	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
35	24, 34	exlimi 1618	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) → (∃!𝑦 𝐴𝐹𝑦 → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧))))
36	35	imp 124	. . . 4 ⊢ ((∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦) → ∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)))
37	18, 36	impbii 126	. . 3 ⊢ (∃𝑧(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑧)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
38	1, 37	bitri 184	. 2 ⊢ (𝑥 ∈ (𝐹‘𝐴) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦))
39	38	abbi2i 2321	1 ⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  {cab 2192   class class class wbr 4048  ‘cfv 5277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3172  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-iota 5238  df-fv 5285

This theorem is referenced by: (None)