Theorem fv2 2828

Description: Alternate definition of function value. Definition 10.11 of [Quine] p. 68.

Hypothesis

Ref	Expression
fv2.1	⊢ A ∈ V

Assertion

Ref	Expression
fv2	⊢ (F ‘A) = ∪{x∣∀y(AFy ↔ y = x)}

Distinct variable group(s):   x,y,A   x,F,y

Proof of Theorem fv2

Step	Hyp	Ref	Expression
1		df-fv 2438	. 2 ⊢ (F ‘A) = ∪{x∣(F “ {A}) = {x}}
2		dfcleq 1098	. . . . 5 ⊢ ((F “ {A}) = {x} ↔ ∀y(y ∈ (F “ {A}) ↔ y ∈ {x}))
3		dfima2 2604	. . . . . . . . . 10 ⊢ (F “ {A}) = {y∣∃x ∈ {A}xFy}
4	3	cleqabi 1176	. . . . . . . . 9 ⊢ (y ∈ (F “ {A}) ↔ ∃x ∈ {A}xFy)
5		df-rex 1206	. . . . . . . . 9 ⊢ (∃x ∈ {A}xFy ↔ ∃x(x ∈ {A} ∧ xFy))
6	4, 5	bitr 151	. . . . . . . 8 ⊢ (y ∈ (F “ {A}) ↔ ∃x(x ∈ {A} ∧ xFy))
7		elsn 1820	. . . . . . . . . 10 ⊢ (x ∈ {A} ↔ x = A)
8	7	anbi1i 368	. . . . . . . . 9 ⊢ ((x ∈ {A} ∧ xFy) ↔ (x = A ∧ xFy))
9	8	biex 733	. . . . . . . 8 ⊢ (∃x(x ∈ {A} ∧ xFy) ↔ ∃x(x = A ∧ xFy))
10		fv2.1	. . . . . . . . 9 ⊢ A ∈ V
11		breq1 2065	. . . . . . . . 9 ⊢ (x = A → (xFy ↔ AFy))
12	10, 11	ceqsexv 1371	. . . . . . . 8 ⊢ (∃x(x = A ∧ xFy) ↔ AFy)
13	6, 9, 12	3bitr 155	. . . . . . 7 ⊢ (y ∈ (F “ {A}) ↔ AFy)
14		elsn 1820	. . . . . . 7 ⊢ (y ∈ {x} ↔ y = x)
15	13, 14	bibi12i 462	. . . . . 6 ⊢ ((y ∈ (F “ {A}) ↔ y ∈ {x}) ↔ (AFy ↔ y = x))
16	15	bial 695	. . . . 5 ⊢ (∀y(y ∈ (F “ {A}) ↔ y ∈ {x}) ↔ ∀y(AFy ↔ y = x))
17	2, 16	bitr 151	. . . 4 ⊢ ((F “ {A}) = {x} ↔ ∀y(AFy ↔ y = x))
18	17	biabi 1181	. . 3 ⊢ {x∣(F “ {A}) = {x}} = {x∣∀y(AFy ↔ y = x)}
19	18	unieqi 1928	. 2 ⊢ ∪{x∣(F “ {A}) = {x}} = ∪{x∣∀y(AFy ↔ y = x)}
20	1, 19	eqtr 1119	1 ⊢ (F ‘A) = ∪{x∣∀y(AFy ↔ y = x)}

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  {csn 1808  ∪cuni 1919   class class class wbr 2054   “ cima 2413   ‘cfv 2422

This theorem is referenced by:  elfv 2830

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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