Theorem fnopabg 2745

Description: Functionality and domain of an ordered pair 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(s):   x,y,A

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . . . . 7 ⊢ (∀x ∈ A ∃!yφ → ∀x∀x ∈ A ∃!yφ)
2		ra4 1243	. . . . . . . 8 ⊢ (∀x ∈ A ∃!yφ → (x ∈ A → ∃!yφ))
3		eumo 1037	. . . . . . . . . 10 ⊢ (∃!yφ → ∃*yφ)
4	3	syl3 18	. . . . . . . . 9 ⊢ ((x ∈ A → ∃!yφ) → (x ∈ A → ∃*yφ))
5		moanimv 1052	. . . . . . . . 9 ⊢ (∃*y(x ∈ A ∧ φ) ↔ (x ∈ A → ∃*yφ))
6	4, 5	sylibr 175	. . . . . . . 8 ⊢ ((x ∈ A → ∃!yφ) → ∃*y(x ∈ A ∧ φ))
7	2, 6	syl 12	. . . . . . 7 ⊢ (∀x ∈ A ∃!yφ → ∃*y(x ∈ A ∧ φ))
8	1, 7	19.21ai 740	. . . . . 6 ⊢ (∀x ∈ A ∃!yφ → ∀x∃*y(x ∈ A ∧ φ))
9		funopab 2694	. . . . . 6 ⊢ (Fun {⟨x, y⟩∣(x ∈ A ∧ φ)} ↔ ∀x∃*y(x ∈ A ∧ φ))
10	8, 9	sylibr 175	. . . . 5 ⊢ (∀x ∈ A ∃!yφ → Fun {⟨x, y⟩∣(x ∈ A ∧ φ)})
11		euex 1021	. . . . . . 7 ⊢ (∃!yφ → ∃yφ)
12	11	r19.20si 1254	. . . . . 6 ⊢ (∀x ∈ A ∃!yφ → ∀x ∈ A ∃yφ)
13		dmopab2 2541	. . . . . 6 ⊢ (∀x ∈ A ∃yφ ↔ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)
14	12, 13	sylib 173	. . . . 5 ⊢ (∀x ∈ A ∃!yφ → dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A)
15	10, 14	jca 236	. . . 4 ⊢ (∀x ∈ A ∃!yφ → (Fun {⟨x, y⟩∣(x ∈ A ∧ φ)} ∧ dom {⟨x, y⟩∣(x ∈ A ∧ φ)} = A))
16		df-fn 2433	. . . 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 175	. . 3 ⊢ (∀x ∈ A ∃!yφ → {⟨x, y⟩∣(x ∈ A ∧ φ)} Fn A)
18		fnopabg.1	. . . 4 ⊢ F = {⟨x, y⟩∣(x ∈ A ∧ φ)}
19		fneq1 2718	. . . 4 ⊢ (F = {⟨x, y⟩∣(x ∈ A ∧ φ)} → (F Fn A ↔ {⟨x, y⟩∣(x ∈ A ∧ φ)} Fn A))
20	18, 19	ax-mp 6	. . 3 ⊢ (F Fn A ↔ {⟨x, y⟩∣(x ∈ A ∧ φ)} Fn A)
21	17, 20	sylibr 175	. 2 ⊢ (∀x ∈ A ∃!yφ → F Fn A)
22		hbopab1 2112	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)} → ∀x z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
23	18	eleq2i 1153	. . . . 5 ⊢ (z ∈ F ↔ z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
24	23	bial 695	. . . . 5 ⊢ (∀x z ∈ F ↔ ∀x z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
25	22, 23, 24	3imtr4 192	. . . 4 ⊢ (z ∈ F → ∀x z ∈ F)
26		ax-17 925	. . . 4 ⊢ (z ∈ A → ∀x z ∈ A)
27	25, 26	hbfn 2720	. . 3 ⊢ (F Fn A → ∀x F Fn A)
28		fneu2 2729	. . . . . 6 ⊢ ((F Fn A ∧ x ∈ A) → ∃!z⟨x, z⟩ ∈ F)
29		ax-17 925	. . . . . . . 8 ⊢ (w ∈ ⟨x, z⟩ → ∀y w ∈ ⟨x, z⟩)
30		hbopab2 2113	. . . . . . . . 9 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)} → ∀y z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
31	23	bial 695	. . . . . . . . 9 ⊢ (∀y z ∈ F ↔ ∀y z ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
32	30, 23, 31	3imtr4 192	. . . . . . . 8 ⊢ (z ∈ F → ∀y z ∈ F)
33	29, 32	hbel 1172	. . . . . . 7 ⊢ (⟨x, z⟩ ∈ F → ∀y⟨x, z⟩ ∈ F)
34		"ax-17.html">ax-17 925	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ F → ∀z⟨x, y⟩ ∈ F)
35		opeq2 1877	. . . . . . . 8 ⊢ (z = y → ⟨x, z⟩ = ⟨x, y⟩)
36	35	eleq1d 1155	. . . . . . 7 ⊢ (z = y → (⟨x, z⟩ ∈ F ↔ ⟨x, y⟩ ∈ F))
37	33, 34, 36	cbveu 1018	. . . . . 6 ⊢ (∃!z⟨x, z⟩ ∈ F ↔ ∃!y⟨x, y⟩ ∈ F)
38	28, 37	sylib 173	. . . . 5 ⊢ ((F Fn A ∧ x ∈ A) → ∃!y⟨x, y⟩ ∈ F)
39	18	eleq2i 1153	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)})
40		opabid 2099	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ A ∧ φ)} ↔ (x ∈ A ∧ φ))
41	39, 40	bitr 151	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ φ))
42	41	bieu 1014	. . . . . . 7 ⊢ (∃!y⟨x, y⟩ ∈ F ↔ ∃!y(x ∈ A ∧ φ))
43		euanv 1053	. . . . . . 7 ⊢ (∃!y(x ∈ A ∧ φ) ↔ (x ∈ A ∧ ∃!yφ))
44	42, 43	bitr 151	. . . . . 6 ⊢ (∃!y⟨x, y⟩ ∈ F ↔ (x ∈ A ∧ ∃!yφ))
45	44	pm3.27bd 263	. . . . 5 ⊢ (∃!y⟨x, y⟩ ∈ F → ∃!yφ)
46	38, 45	syl 12	. . . 4 ⊢ ((F Fn A ∧ x ∈ A) → ∃!yφ)
47	46	exp 291	. . 3 ⊢ (F Fn A → (x ∈ A → ∃!yφ))
48	27, 47	r19.21ai 1258	. 2 ⊢ (F Fn A → ∀x ∈ A ∃!yφ)
49	21, 48	impbi 139	1 ⊢ (∀x ∈ A ∃!yφ ↔ F Fn A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ⟨cop 1810  {copab 2055  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fnopab 2746  fopab2 2891  en2d 3303

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-ral 1205  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-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433

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