Theorem funpartlem 32364

Description: Lemma for funpartfun 32365. 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 3402	. 2 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2		vsnid 4397	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
3		eleq2 2870	. . . . 5 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
4	2, 3	mpbiri 249	. . . 4 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5		n0i 4115	. . . . 5 ⊢ (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6		snprc 4438	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 207	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8	7	imaeq2d 5670	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9		ima0 5685	. . . . . 6 ⊢ (𝐹 “ ∅) = ∅
10	8, 9	syl6eq 2852	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
11	5, 10	nsyl2 144	. . . 4 ⊢ (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
12	4, 11	syl 17	. . 3 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
13	12	exlimiv 2020	. 2 ⊢ (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14		eleq1 2869	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15		sneq 4374	. . . . . 6 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
16	15	imaeq2d 5670	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
17	16	eqeq1d 2804	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
18	17	exbidv 2012	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19		vex 3390	. . . . 5 ⊢ 𝑦 ∈ V
20	19	eldm 5516	. . . 4 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21		brxp 5341	. . . . . . . . . 10 ⊢ (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 ∈ Singletons ))
22	19, 21	mpbiran 691	. . . . . . . . 9 ⊢ (𝑦(V × Singletons )𝑧 ↔ 𝑧 ∈ Singletons )
23		elsingles 32340	. . . . . . . . 9 ⊢ (𝑧 ∈ Singletons ↔ ∃𝑥 𝑧 = {𝑥})
24	22, 23	bitri 266	. . . . . . . 8 ⊢ (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
25	24	anbi2i 611	. . . . . . 7 ⊢ ((𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26		brin 4889	. . . . . . 7 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧))
27		19.42v 2043	. . . . . . 7 ⊢ (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
28	25, 26, 27	3bitr4i 294	. . . . . 6 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
29	28	exbii 1933	. . . . 5 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
30		excom 2208	. . . . 5 ⊢ (∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
31	29, 30	bitri 266	. . . 4 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
32		exancom 1947	. . . . . 6 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33		snex 5092	. . . . . . 7 ⊢ {𝑥} ∈ V
34		breq2 4841	. . . . . . 7 ⊢ (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧 ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥}))
35	33, 34	ceqsexv 3432	. . . . . 6 ⊢ (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
36	19, 33	brco 5488	. . . . . . 7 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}))
37		vex 3390	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
38	19, 37	brsingle 32339	. . . . . . . . 9 ⊢ (𝑦Singleton𝑧 ↔ 𝑧 = {𝑦})
39	38	anbi1i 612	. . . . . . . 8 ⊢ ((𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
40	39	exbii 1933	. . . . . . 7 ⊢ (∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41		snex 5092	. . . . . . . . 9 ⊢ {𝑦} ∈ V
42		breq1 4840	. . . . . . . . 9 ⊢ (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
43	41, 42	ceqsexv 3432	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
44	41, 33	brimage 32348	. . . . . . . 8 ⊢ ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45		eqcom 2809	. . . . . . . 8 ⊢ ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
46	43, 44, 45	3bitri 288	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
47	36, 40, 46	3bitri 288	. . . . . 6 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
48	32, 35, 47	3bitri 288	. . . . 5 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
49	48	exbii 1933	. . . 4 ⊢ (∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
50	20, 31, 49	3bitri 288	. . 3 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
51	14, 18, 50	vtoclbg 3456	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
52	1, 13, 51	pm5.21nii 369	1 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Syntax hints:  ¬ wn 3   ↔ wb 197   ∧ wa 384   = wceq 1637  ∃wex 1859   ∈ wcel 2155  Vcvv 3387   ∩ cin 3762  ∅c0 4110  {csn 4364   class class class wbr 4837   × cxp 5303  dom cdm 5305   “ cima 5308   ∘ ccom 5309  Singletoncsingle 32260   Singletons csingles 32261  Imagecimage 32262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-symdif 4036  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-eprel 5218  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-fo 6101  df-fv 6103  df-1st 7392  df-2nd 7393  df-txp 32276  df-singleton 32284  df-singles 32285  df-image 32286

This theorem is referenced by:  funpartfun  32365  funpartfv  32367