Theorem fnoprab 3038

Description: Functionality and domain of an operation abstraction.

Hypothesis

Ref	Expression
fnoprab.1	⊢ (φ → ∃!zψ)

Assertion

Ref	Expression
fnoprab	⊢ {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} Fn {⟨x, y⟩∣φ}

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

Proof of Theorem fnoprab

Step	Hyp	Ref	Expression
1		fnoprab.1	. . . . . 6 ⊢ (φ → ∃!zψ)
2		eumo 1037	. . . . . 6 ⊢ (∃!zψ → ∃*zψ)
3	1, 2	syl 12	. . . . 5 ⊢ (φ → ∃*zψ)
4		moanimv 1052	. . . . 5 ⊢ (∃*z(φ ∧ ψ) ↔ (φ → ∃*zψ))
5	3, 4	mpbir 165	. . . 4 ⊢ ∃*z(φ ∧ ψ)
6	5	funoprab 3037	. . 3 ⊢ Fun {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)}
7		dmoprab 3031	. . . 4 ⊢ dom {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} = {⟨x, y⟩∣∃z(φ ∧ ψ)}
8		pm3.26 256	. . . . . . 7 ⊢ ((φ ∧ ψ) → φ)
9	8	19.23aiv 952	. . . . . 6 ⊢ (∃z(φ ∧ ψ) → φ)
10		euex 1021	. . . . . . . . 9 ⊢ (∃!zψ → ∃zψ)
11	1, 10	syl 12	. . . . . . . 8 ⊢ (φ → ∃zψ)
12	11	ancli 244	. . . . . . 7 ⊢ (φ → (φ ∧ ∃zψ))
13		19.42v 966	. . . . . . 7 ⊢ (∃z(φ ∧ ψ) ↔ (φ ∧ ∃zψ))
14	12, 13	sylibr 175	. . . . . 6 ⊢ (φ → ∃z(φ ∧ ψ))
15	9, 14	impbi 139	. . . . 5 ⊢ (∃z(φ ∧ ψ) ↔ φ)
16	15	biopabi 2103	. . . 4 ⊢ {⟨x, y⟩∣∃z(φ ∧ ψ)} = {⟨x, y⟩∣φ}
17	7, 16	eqtr 1119	. . 3 ⊢ dom {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} = {⟨x, y⟩∣φ}
18	6, 17	pm3.2i 234	. 2 ⊢ (Fun {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} ∧ dom {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} = {⟨x, y⟩∣φ})
19		df-fn 2433	. 2 ⊢ ({⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} Fn {⟨x, y⟩∣φ} ↔ (Fun {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} ∧ dom {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} = {⟨x, y⟩∣φ}))
20	18, 19	mpbir 165	1 ⊢ {⟨⟨x, y⟩, z⟩∣(φ ∧ ψ)} Fn {⟨x, y⟩∣φ}

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008   = wceq 1091  {copab 2055  dom cdm 2410  Fun wfun 2416   Fn wfn 2417  {copab2 3002

This theorem is referenced by:  fnoprab2 3039  oprabval 3047

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-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433  df-oprab 3004

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