Theorem funpartlem 36124

Description: Lemma for funpartfun 36125. 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 3450	. 2 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2		vsnid 4607	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
3		eleq2 2825	. . . . 5 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
4	2, 3	mpbiri 258	. . . 4 ⊢ ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5		n0i 4280	. . . . 5 ⊢ (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6		snprc 4661	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
7	6	biimpi 216	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
8	7	imaeq2d 6025	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9		ima0 6042	. . . . . 6 ⊢ (𝐹 “ ∅) = ∅
10	8, 9	eqtrdi 2787	. . . . 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 1932	. 2 ⊢ (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14		eleq1 2824	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15		sneq 4577	. . . . . 6 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴})
16	15	imaeq2d 6025	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
17	16	eqeq1d 2738	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
18	17	exbidv 1923	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19		vex 3433	. . . . 5 ⊢ 𝑦 ∈ V
20	19	eldm 5855	. . . 4 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21		brxp 5680	. . . . . . . . . 10 ⊢ (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 ∈ Singletons ))
22	19, 21	mpbiran 710	. . . . . . . . 9 ⊢ (𝑦(V × Singletons )𝑧 ↔ 𝑧 ∈ Singletons )
23		elsingles 36098	. . . . . . . . 9 ⊢ (𝑧 ∈ Singletons ↔ ∃𝑥 𝑧 = {𝑥})
24	22, 23	bitri 275	. . . . . . . 8 ⊢ (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
25	24	anbi2i 624	. . . . . . 7 ⊢ ((𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26		brin 5137	. . . . . . 7 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑦(V × Singletons )𝑧))
27		19.42v 1955	. . . . . . 7 ⊢ (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
28	25, 26, 27	3bitr4i 303	. . . . . 6 ⊢ (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
29	28	exbii 1850	. . . . 5 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
30		excom 2168	. . . . 5 ⊢ (∃𝑧∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
31	29, 30	bitri 275	. . . 4 ⊢ (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}))
32		exancom 1863	. . . . . 6 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33		vsnex 5377	. . . . . . 7 ⊢ {𝑥} ∈ V
34		breq2 5089	. . . . . . 7 ⊢ (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧 ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥}))
35	33, 34	ceqsexv 3478	. . . . . 6 ⊢ (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
36	19, 33	brco 5825	. . . . . . 7 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}))
37		vex 3433	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
38	19, 37	brsingle 36097	. . . . . . . . 9 ⊢ (𝑦Singleton𝑧 ↔ 𝑧 = {𝑦})
39	38	anbi1i 625	. . . . . . . 8 ⊢ ((𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
40	39	exbii 1850	. . . . . . 7 ⊢ (∃𝑧(𝑦Singleton𝑧 ∧ 𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41		vsnex 5377	. . . . . . . . 9 ⊢ {𝑦} ∈ V
42		breq1 5088	. . . . . . . . 9 ⊢ (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
43	41, 42	ceqsexv 3478	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
44	41, 33	brimage 36106	. . . . . . . 8 ⊢ ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45		eqcom 2743	. . . . . . . 8 ⊢ ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
46	43, 44, 45	3bitri 297	. . . . . . 7 ⊢ (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
47	36, 40, 46	3bitri 297	. . . . . 6 ⊢ (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
48	32, 35, 47	3bitri 297	. . . . 5 ⊢ (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
49	48	exbii 1850	. . . 4 ⊢ (∃𝑥∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ 𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
50	20, 31, 49	3bitri 297	. . 3 ⊢ (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
51	14, 18, 50	vtoclbg 3502	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
52	1, 13, 51	pm5.21nii 378	1 ⊢ (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  ∅c0 4273  {csn 4567   class class class wbr 5085   × cxp 5629  dom cdm 5631   “ cima 5634   ∘ ccom 5635  Singletoncsingle 36018   Singletons csingles 36019  Imagecimage 36020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-symdif 4193  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-1st 7942  df-2nd 7943  df-txp 36034  df-singleton 36042  df-singles 36043  df-image 36044

This theorem is referenced by:  funpartfun  36125  funpartfv  36127