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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.
StepHypRef 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}))
42, 3mpbiri 249 . . . 4 ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5 n0i 4115 . . . . 5 (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6 snprc 4438 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 207 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
87imaeq2d 5670 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9 ima0 5685 . . . . . 6 (𝐹 “ ∅) = ∅
108, 9syl6eq 2852 . . . . 5 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
115, 10nsyl2 144 . . . 4 (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
124, 11syl 17 . . 3 ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
1312exlimiv 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})
1615imaeq2d 5670 . . . . 5 (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
1716eqeq1d 2804 . . . 4 (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
1817exbidv 2012 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19 vex 3390 . . . . 5 𝑦 ∈ V
2019eldm 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 ))
2219, 21mpbiran 691 . . . . . . . . 9 (𝑦(V × Singletons )𝑧𝑧 Singletons )
23 elsingles 32340 . . . . . . . . 9 (𝑧 Singletons ↔ ∃𝑥 𝑧 = {𝑥})
2422, 23bitri 266 . . . . . . . 8 (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
2524anbi2i 611 . . . . . . 7 ((𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26 brin 4889 . . . . . . 7 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧))
27 19.42v 2043 . . . . . . 7 (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
2825, 26, 273bitr4i 294 . . . . . 6 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
2928exbii 1933 . . . . 5 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
30 excom 2208 . . . . 5 (∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
3129, 30bitri 266 . . . 4 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
32 exancom 1947 . . . . . 6 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33 snex 5092 . . . . . . 7 {𝑥} ∈ V
34 breq2 4841 . . . . . . 7 (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(Image𝐹 ∘ Singleton){𝑥}))
3533, 34ceqsexv 3432 . . . . . 6 (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
3619, 33brco 5488 . . . . . . 7 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}))
37 vex 3390 . . . . . . . . . 10 𝑧 ∈ V
3819, 37brsingle 32339 . . . . . . . . 9 (𝑦Singleton𝑧𝑧 = {𝑦})
3938anbi1i 612 . . . . . . . 8 ((𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
4039exbii 1933 . . . . . . 7 (∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41 snex 5092 . . . . . . . . 9 {𝑦} ∈ V
42 breq1 4840 . . . . . . . . 9 (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
4341, 42ceqsexv 3432 . . . . . . . 8 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
4441, 33brimage 32348 . . . . . . . 8 ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45 eqcom 2809 . . . . . . . 8 ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4643, 44, 453bitri 288 . . . . . . 7 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4736, 40, 463bitri 288 . . . . . 6 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
4832, 35, 473bitri 288 . . . . 5 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4948exbii 1933 . . . 4 (∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5020, 31, 493bitri 288 . . 3 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5114, 18, 50vtoclbg 3456 . 2 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
521, 13, 51pm5.21nii 369 1 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Colors of variables: wff setvar class
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
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