Theorem dffunmof 2678

Description: Function definition requiring only that x and y not be "free" in A (but not necessarily absent from it), using "at most one" notation.

Hypotheses

Ref	Expression
dffunmof.1	⊢ (z ∈ A → ∀x z ∈ A)
dffunmof.2	⊢ (z ∈ A → ∀y z ∈ A)

Assertion

Ref	Expression
dffunmof	⊢ (Fun A ↔ (Rel A ∧ ∀x∃*y xAy))

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

Proof of Theorem dffunmof

Step	Hyp	Ref	Expression
1		dffun3 2675	. 2 ⊢ (Fun A ↔ (Rel A ∧ ∀w∃u∀v(wAv → v = u)))
2		ax-17 925	. . . . . . 7 ⊢ (z ∈ w → ∀y z ∈ w)
3		dffunmof.2	. . . . . . 7 ⊢ (z ∈ A → ∀y z ∈ A)
4		ax-17 925	. . . . . . 7 ⊢ (z ∈ v → ∀y z ∈ v)
5	2, 3, 4	hbbr 2095	. . . . . 6 ⊢ (wAv → ∀y wAv)
6		ax-17 925	. . . . . 6 ⊢ (wAy → ∀v wAy)
7		breq2 2066	. . . . . 6 ⊢ (v = y → (wAv ↔ wAy))
8	5, 6, 7	cbvmo 1034	. . . . 5 ⊢ (∃*v wAv ↔ ∃*y wAy)
9	8	bial 695	. . . 4 ⊢ (∀w∃*v wAv ↔ ∀w∃*y wAy)
10		ax-17 925	. . . . . 6 ⊢ (wAv → ∀u wAv)
11	10	mo2 1026	. . . . 5 ⊢ (∃*v wAv ↔ ∃u∀v(wAv → v = u))
12	11	bial 695	. . . 4 ⊢ (∀w∃*v wAv ↔ ∀w∃u∀v(wAv → v = u))
13		ax-17 925	. . . . . . 7 ⊢ (z ∈ w → ∀x z ∈ w)
14		dffunmof.1	. . . . . . 7 ⊢ (z ∈ A → ∀x z ∈ A)
15		ax-17 925	. . . . . . 7 ⊢ (z ∈ y → ∀x z ∈ y)
16	13, 14, 15	hbbr 2095	. . . . . 6 ⊢ (wAy → ∀x wAy)
17	16	hbmo 1033	. . . . 5 ⊢ (∃*y wAy → ∀x∃*y wAy)
18		ax-17 925	. . . . 5 ⊢ (∃*y xAy → ∀w∃*y xAy)
19		ax-17 925	. . . . . 6 ⊢ (w = x → ∀y w = x)
20		breq1 2065	. . . . . 6 ⊢ (w = x → (wAy ↔ xAy))
21	19, 20	bimod 1030	. . . . 5 ⊢ (w = x → (∃*y wAy ↔ ∃*y xAy))
22	17, 18, 21	cbval 848	. . . 4 ⊢ (∀w∃*y wAy ↔ ∀x∃*y xAy)
23	9, 12, 22	3bitr3r 157	. . 3 ⊢ (∀x∃*y xAy ↔ ∀w∃u∀v(wAv → v = u))
24	23	anbi2i 367	. 2 ⊢ ((Rel A ∧ ∀x∃*y xAy) ↔ (Rel A ∧ ∀w∃u∀v(wAv → v = u)))
25	1, 24	bitr4 154	1 ⊢ (Fun A ↔ (Rel A ∧ ∀x∃*y xAy))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  ∃*wmo 1008   ∈ wcel 1092   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  dffunmo 2679  funopab 2694

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-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  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-br 2063  df-opab 2098  df-id 2125  df-cnv 2426  df-co 2427  df-fun 2432

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