Theorem funpartlem 34171

Description: Lemma for funpartfun 34172. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
funpartlem	⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem funpartlem

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2		vsnid 4595	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
3		eleq2 2827	. . . . 5 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
4	2, 3	mpbiri 257	. . . 4 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5		n0i 4264	. . . . 5 ⊢ (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6		snprc 4650	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8	7	imaeq2d 5958	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9		ima0 5974	. . . . . 6 ⊢ (𝐹 “ ∅) = ∅
10	8, 9	eqtrdi 2795	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
11	5, 10	nsyl2 141	. . . 4 ⊢ (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
12	4, 11	syl 17	. . 3 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
13	12	exlimiv 1934	. 2 ⊢ (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14		eleq1 2826	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15		sneq 4568	. . . . . 6 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
16	15	imaeq2d 5958	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
17	16	eqeq1d 2740	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
18	17	exbidv 1925	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
20	19	eldm 5798	. . . 4 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21		brxp 5627	. . . . . . . . . 10 ⊢ (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 ∈ Singletons ))
22	19, 21	mpbiran 705	. . . . . . . . 9 ⊢ (𝑦(V × Singletons )𝑧 ↔ 𝑧 ∈ Singletons )
23		elsingles 34147	. . . . . . . . 9 ⊢ (𝑧 ∈ Singletons ↔ ∃𝑥 𝑧 = {𝑥})
24	22, 23	bitri 274	. . . . . . . 8 ⊢ (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
25	24	anbi2i 622	. . . . . . 7 ⊢ ((𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26		brin 5122	. . . . . . 7 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧))
27		19.42v 1958	. . . . . . 7 ⊢ (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
28	25, 26, 27	3bitr4i 302	. . . . . 6 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
29	28	exbii 1851	. . . . 5 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
30		excom 2164	. . . . 5 ⊢ (∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
31	29, 30	bitri 274	. . . 4 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
32		exancom 1865	. . . . . 6 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33		snex 5349	. . . . . . 7 ⊢ {𝑥} ∈ V
34		breq2 5074	. . . . . . 7 ⊢ (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧 ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥}))
35	33, 34	ceqsexv 3469	. . . . . 6 ⊢ (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
36	19, 33	brco 5768	. . . . . . 7 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}))
37		vex 3426	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
38	19, 37	brsingle 34146	. . . . . . . . 9 ⊢ (𝑦Singleton𝑧 ↔ 𝑧 = {𝑦})
39	38	anbi1i 623	. . . . . . . 8 ⊢ ((𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
40	39	exbii 1851	. . . . . . 7 ⊢ (∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41		snex 5349	. . . . . . . . 9 ⊢ {𝑦} ∈ V
42		breq1 5073	. . . . . . . . 9 ⊢ (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
43	41, 42	ceqsexv 3469	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
44	41, 33	brimage 34155	. . . . . . . 8 ⊢ ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45		eqcom 2745	. . . . . . . 8 ⊢ ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
46	43, 44, 45	3bitri 296	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
47	36, 40, 46	3bitri 296	. . . . . 6 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
48	32, 35, 47	3bitri 296	. . . . 5 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
49	48	exbii 1851	. . . 4 ⊢ (∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
50	20, 31, 49	3bitri 296	. . 3 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
51	14, 18, 50	vtoclbg 3497	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
52	1, 13, 51	pm5.21nii 379	1 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  ∅c0 4253  {csn 4558   class class class wbr 5070   × cxp 5578  dom cdm 5580   “ cima 5583   ∘ ccom 5584  Singletoncsingle 34067   Singletons csingles 34068  Imagecimage 34069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-symdif 4173  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-eprel 5486  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-1st 7804  df-2nd 7805  df-txp 34083  df-singleton 34091  df-singles 34092  df-image 34093

This theorem is referenced by:  funpartfun  34172  funpartfv  34174