Theorem fnopabg 5399

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 2129	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
2	1	albii 1493	. . . . 5 ⊢ (∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
3		funopab 5306	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-ral 2489	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦𝜑))
5	2, 3, 4	3bitr4ri 213	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
6		dmopab3 4891	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
7	5, 6	anbi12i 460	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) ↔ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴))
8		r19.26 2632	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑))
9		df-fn 5274	. . 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 212	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑) ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
11		eu5 2101	. . . 4 ⊢ (∃!𝑦𝜑 ↔ (∃𝑦𝜑 ∧ ∃*𝑦𝜑))
12		ancom 266	. . . 4 ⊢ ((∃𝑦𝜑 ∧ ∃*𝑦𝜑) ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
13	11, 12	bitri 184	. . 3 ⊢ (∃!𝑦𝜑 ↔ (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
14	13	ralbii 2512	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ 𝐴 (∃*𝑦𝜑 ∧ ∃𝑦𝜑))
15		fnopabg.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
16	15	fneq1i 5368	. 2 ⊢ (𝐹 Fn 𝐴 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} Fn 𝐴)
17	10, 14, 16	3bitr4i 212	1 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1515  ∃!weu 2054  ∃*wmo 2055   ∈ wcel 2176  ∀wral 2484  {copab 4104  dom cdm 4675  Fun wfun 5265   Fn wfn 5266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-fun 5273  df-fn 5274

This theorem is referenced by:  fnopab  5400  mptfng  5401  uchoice  6223