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Theorem funpartlem 34241
Description: Lemma for funpartfun 34242. 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 3449 . 2 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2 vsnid 4600 . . . . 5 𝑥 ∈ {𝑥}
3 eleq2 2827 . . . . 5 ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
42, 3mpbiri 257 . . . 4 ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5 n0i 4269 . . . . 5 (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6 snprc 4655 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
87imaeq2d 5971 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9 ima0 5987 . . . . . 6 (𝐹 “ ∅) = ∅
108, 9eqtrdi 2794 . . . . 5 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
115, 10nsyl2 141 . . . 4 (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
124, 11syl 17 . . 3 ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
1312exlimiv 1933 . 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 4573 . . . . . 6 (𝑦 = 𝐴 → {𝑦} = {𝐴})
1615imaeq2d 5971 . . . . 5 (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
1716eqeq1d 2740 . . . 4 (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
1817exbidv 1924 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19 vex 3435 . . . . 5 𝑦 ∈ V
2019eldm 5811 . . . 4 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21 brxp 5638 . . . . . . . . . 10 (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 Singletons ))
2219, 21mpbiran 706 . . . . . . . . 9 (𝑦(V × Singletons )𝑧𝑧 Singletons )
23 elsingles 34217 . . . . . . . . 9 (𝑧 Singletons ↔ ∃𝑥 𝑧 = {𝑥})
2422, 23bitri 274 . . . . . . . 8 (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
2524anbi2i 623 . . . . . . 7 ((𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26 brin 5128 . . . . . . 7 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧))
27 19.42v 1957 . . . . . . 7 (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
2825, 26, 273bitr4i 303 . . . . . 6 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
2928exbii 1850 . . . . 5 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
30 excom 2162 . . . . 5 (∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
3129, 30bitri 274 . . . 4 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
32 exancom 1864 . . . . . 6 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33 snex 5356 . . . . . . 7 {𝑥} ∈ V
34 breq2 5080 . . . . . . 7 (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(Image𝐹 ∘ Singleton){𝑥}))
3533, 34ceqsexv 3478 . . . . . 6 (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
3619, 33brco 5781 . . . . . . 7 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}))
37 vex 3435 . . . . . . . . . 10 𝑧 ∈ V
3819, 37brsingle 34216 . . . . . . . . 9 (𝑦Singleton𝑧𝑧 = {𝑦})
3938anbi1i 624 . . . . . . . 8 ((𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
4039exbii 1850 . . . . . . 7 (∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41 snex 5356 . . . . . . . . 9 {𝑦} ∈ V
42 breq1 5079 . . . . . . . . 9 (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
4341, 42ceqsexv 3478 . . . . . . . 8 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
4441, 33brimage 34225 . . . . . . . 8 ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45 eqcom 2745 . . . . . . . 8 ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4643, 44, 453bitri 297 . . . . . . 7 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4736, 40, 463bitri 297 . . . . . 6 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
4832, 35, 473bitri 297 . . . . 5 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4948exbii 1850 . . . 4 (∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5020, 31, 493bitri 297 . . 3 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5114, 18, 50vtoclbg 3506 . 2 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
521, 13, 51pm5.21nii 380 1 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  Vcvv 3431  cin 3887  c0 4258  {csn 4563   class class class wbr 5076   × cxp 5589  dom cdm 5591  cima 5594  ccom 5595  Singletoncsingle 34137   Singletons csingles 34138  Imagecimage 34139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-symdif 4178  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-eprel 5497  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-fo 6441  df-fv 6443  df-1st 7831  df-2nd 7832  df-txp 34153  df-singleton 34161  df-singles 34162  df-image 34163
This theorem is referenced by:  funpartfun  34242  funpartfv  34244
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