Theorem fnopabg 5311

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		moanimv 2089	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
2	1	albii 1458	. . . . 5 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
3		funopab 5223	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-ral 2449	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
5	2, 3, 4	3bitr4ri 212	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		dmopab3 4817	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
7	5, 6	anbi12i 456	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴))
8		r19.26 2592	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑))
9		df-fn 5191	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴 ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴))
10	7, 8, 9	3bitr4i 211	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
11		eu5 2061	. . . 4 ⊢ (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12		ancom 264	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃*𝑦𝜑) ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
13	11, 12	bitri 183	. . 3 ⊢ (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
14	13	ralbii 2472	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
15		fnopabg.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
16	15	fneq1i 5282	. 2 ⊢ (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
17	10, 14, 16	3bitr4i 211	1 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480  ∃!weu 2014  ∃*wmo 2015   ∈ wcel 2136  ∀wral 2444  {copab 4042  dom cdm 4604  Fun wfun 5182   Fn wfn 5183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  fnopab  5312  mptfng  5313