Theorem fnopabg 3601

Description: Functionality and domain of an ordered-pair class abstraction.

Hypothesis

Ref	Expression
fnopabg.1	⊢ F = {⟨x, y⟩∣(x ∈ A ⋀ φ)}

Assertion

Ref	Expression
fnopabg	⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)

Distinct variable group:   x,y,A

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		hbra1 1679	. . . . . . 7 ⊢ (∀x ∈ A ∃!yφ → ∀x∀x ∈ A ∃!yφ)
2		ra4 1686	. . . . . . . 8 ⊢ (∀x ∈ A ∃!yφ → (x ∈ A → ∃!yφ))
3		eumo 1404	. . . . . . . . . 10 ⊢ (∃!yφ → ∃*yφ)
4	3	imim2i 17	. . . . . . . . 9 ⊢ ((x ∈ A → ∃!yφ) → (x ∈ A → ∃*yφ))
5		moanimv 1422	. . . . . . . . 9 ⊢ (∃*y(x ∈ A ⋀ φ) ↔ (x ∈ A → ∃*yφ))
6	4, 5	sylibr 200	. . . . . . . 8 ⊢ ((x ∈ A → ∃!yφ) → ∃*y(x ∈ A ⋀ φ))
7	2, 6	syl 10	. . . . . . 7 ⊢ (∀x ∈ A ∃!yφ → ∃*y(x ∈ A ⋀ φ))
8	1, 7	19.21ai 995	. . . . . 6 ⊢ (∀x ∈ A ∃!yφ → ∀x∃*y(x ∈ A ⋀ φ))
9		funopab 3534	. . . . . 6 ⊢ (Fun {⟨x, y⟩∣(x ∈ A ⋀ φ)} ↔ ∀x∃*y(x ∈ A ⋀ φ))
10	8, 9	sylibr 200	. . . . 5 ⊢ (∀x ∈ A ∃!yφ → Fun {⟨x, y⟩∣(x ∈ A ⋀ φ)})
11		euex 1387	. . . . . . 7 ⊢ (∃!yφ → ∃yφ)
12	11	r19.20si 1698	. . . . . 6 ⊢ (∀x ∈ A ∃!yφ → ∀x ∈ A ∃yφ)
13		dmopab3 3311	. . . . . 6 ⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A)
14	12, 13	sylib 198	. . . . 5 ⊢ (∀x ∈ A ∃!yφ → dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A)
15	10, 14	jca 288	. . . 4 ⊢ (∀x ∈ A ∃!yφ → (Fun {⟨x, y⟩∣(x ∈ A ⋀ φ)} ⋀ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A))
16		df-fn 3183	. . . 4 ⊢ ({⟨x, y⟩∣(x ∈ A ⋀ φ)} Fn A ↔ (Fun {⟨x, y⟩∣(x ∈ A ⋀ φ)} ⋀ dom {⟨x, y⟩∣(x ∈ A ⋀ φ)} = A))
17	15, 16	sylibr 200	. . 3 ⊢ (∀x ∈ A ∃!yφ → {⟨x, y⟩∣(x ∈ A ⋀ φ)} Fn A)
18		fnopabg.1	. . . 4 ⊢ F = {⟨x, y⟩∣(x ∈ A ⋀ φ)}
19		fneq1 3568	. . . 4 ⊢ (F = {⟨x, y⟩∣(x ∈ A ⋀ φ)} → (F Fn A ↔ {⟨x, y⟩∣(x ∈ A ⋀ φ)} Fn A))
20	18, 19	ax-mp 7	. . 3 ⊢ (F Fn A ↔ {⟨x, y⟩∣(x ∈ A ⋀ φ)} Fn A)
21	17, 20	sylibr 200	. 2 ⊢ (∀x ∈ A ∃!yφ → F Fn A)
22		hbopab1 2802	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)} → ∀x z ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)})
23	18, 22	hbxfr 1555	. . . 4 ⊢ (z ∈ F → ∀x z ∈ F)
24		ax-17 968	. . . 4 ⊢ (z ∈ A → ∀x z ∈ A)
25	23, 24	hbfn 3570	. . 3 ⊢ (F Fn A → ∀x F Fn A)
26		fneu2 3579	. . . . . 6 ⊢ ((F Fn A ⋀ x ∈ A) → ∃!z⟨x, z⟩ ∈ F)
27		ax-17 968	. . . . . . . 8 ⊢ (w ∈ ⟨x, z⟩ → ∀y w ∈ ⟨x, z⟩)
28		hbopab2 2803	. . . . . . . . 9 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)} → ∀y z ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)})
29	18, 28	hbxfr 1555	. . . . . . . 8 ⊢ (z ∈ F → ∀y z ∈ F)
30	27, 29	hbel 1558	. . . . . . 7 ⊢ (⟨x, z⟩ ∈ F → ∀y⟨x, z⟩ ∈ F)
31		ax-17 968	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ F → ∀z⟨x, y⟩ ∈ F)
32		opeq2 2479	. . . . . . . 8 ⊢ (z = y → ⟨x, z⟩ = ⟨x, y⟩)
33	32	eleq1d 1532	. . . . . . 7 ⊢ (z = y → (⟨x, z⟩ ∈ F ↔ ⟨x, y⟩ ∈ F))
34	30, 31, 33	cbveu 1384	. . . . . 6 ⊢ (∃!z⟨x, z⟩ ∈ F ↔ ∃!y⟨x, y⟩ ∈ F)
35	26, 34	sylib 198	. . . . 5 ⊢ ((F Fn A ⋀ x ∈ A) → ∃!y⟨x, y⟩ ∈ F)
36	18	eleq2i 1530	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)})
37		opabid 2799	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ⋀ φ)} ↔ (x ∈ A ⋀ φ))
38	36, 37	bitr 173	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ F ↔ (x ∈ A ⋀ φ))
39	38	eubii 1380	. . . . . . 7 ⊢ (∃!y⟨x, y⟩ ∈ F ↔ ∃!y(x ∈ A ⋀ φ))
40		euanv 1425	. . . . . . 7 ⊢ (∃!y(x ∈ A ⋀ φ) ↔ (x ∈ A ⋀ ∃!yφ))
41	39, 40	bitr 173	. . . . . 6 ⊢ (∃!y⟨x, y⟩ ∈ F ↔ (x ∈ A ⋀ ∃!yφ))
42	41	pm3.27bi 326	. . . . 5 ⊢ (∃!y⟨x, y⟩ ∈ F → ∃!yφ)
43	35, 42	syl 10	. . . 4 ⊢ ((F Fn A ⋀ x ∈ A) → ∃!yφ)
44	43	ex 373	. . 3 ⊢ (F Fn A → (x ∈ A → ∃!yφ))
45	25, 44	r19.21ai 1704	. 2 ⊢ (F Fn A → ∀x ∈ A ∃!yφ)
46	21, 45	impbi 157	1 ⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 951   = wceq 953   ∈ wcel 955  ∃wex 977  ∃!weu 1373  ∃*wmo 1374  ∀wral 1637  ⟨cop 2401  {copab 2656  dom cdm 3160  Fun wfun 3166   Fn wfn 3167

This theorem is referenced by:  fnopab2g 3602  fnopab 3603  elrnopabg 3785  fopab2 3808  en2d 4381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183

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