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Theorem funpartlem 32174
Description: Lemma for funpartfun 32175. 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 3243 . 2 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2 vsnid 4242 . . . . 5 𝑥 ∈ {𝑥}
3 eleq2 2719 . . . . 5 ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
42, 3mpbiri 248 . . . 4 ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5 n0i 3953 . . . . 5 (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6 snprc 4285 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 206 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
87imaeq2d 5501 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9 ima0 5516 . . . . . 6 (𝐹 “ ∅) = ∅
108, 9syl6eq 2701 . . . . 5 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
115, 10nsyl2 142 . . . 4 (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
124, 11syl 17 . . 3 ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
1312exlimiv 1898 . 2 (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14 eleq1 2718 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15 sneq 4220 . . . . . 6 (𝑦 = 𝐴 → {𝑦} = {𝐴})
1615imaeq2d 5501 . . . . 5 (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
1716eqeq1d 2653 . . . 4 (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
1817exbidv 1890 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19 vex 3234 . . . . 5 𝑦 ∈ V
2019eldm 5353 . . . 4 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21 brxp 5181 . . . . . . . . . 10 (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 Singletons ))
2219, 21mpbiran 973 . . . . . . . . 9 (𝑦(V × Singletons )𝑧𝑧 Singletons )
23 elsingles 32150 . . . . . . . . 9 (𝑧 Singletons ↔ ∃𝑥 𝑧 = {𝑥})
2422, 23bitri 264 . . . . . . . 8 (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
2524anbi2i 730 . . . . . . 7 ((𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26 brin 4737 . . . . . . 7 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧))
27 19.42v 1921 . . . . . . 7 (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
2825, 26, 273bitr4i 292 . . . . . 6 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
2928exbii 1814 . . . . 5 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
30 excom 2082 . . . . 5 (∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
3129, 30bitri 264 . . . 4 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
32 exancom 1827 . . . . . 6 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33 snex 4938 . . . . . . 7 {𝑥} ∈ V
34 breq2 4689 . . . . . . 7 (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(Image𝐹 ∘ Singleton){𝑥}))
3533, 34ceqsexv 3273 . . . . . 6 (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
3619, 33brco 5325 . . . . . . 7 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}))
37 vex 3234 . . . . . . . . . 10 𝑧 ∈ V
3819, 37brsingle 32149 . . . . . . . . 9 (𝑦Singleton𝑧𝑧 = {𝑦})
3938anbi1i 731 . . . . . . . 8 ((𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
4039exbii 1814 . . . . . . 7 (∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41 snex 4938 . . . . . . . . 9 {𝑦} ∈ V
42 breq1 4688 . . . . . . . . 9 (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
4341, 42ceqsexv 3273 . . . . . . . 8 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
4441, 33brimage 32158 . . . . . . . 8 ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45 eqcom 2658 . . . . . . . 8 ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4643, 44, 453bitri 286 . . . . . . 7 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4736, 40, 463bitri 286 . . . . . 6 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
4832, 35, 473bitri 286 . . . . 5 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4948exbii 1814 . . . 4 (∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5020, 31, 493bitri 286 . . 3 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5114, 18, 50vtoclbg 3298 . 2 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
521, 13, 51pm5.21nii 367 1 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cin 3606  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  dom cdm 5143  cima 5146  ccom 5147  Singletoncsingle 32070   Singletons csingles 32071  Imagecimage 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-singleton 32094  df-singles 32095  df-image 32096
This theorem is referenced by:  funpartfun  32175  funpartfv  32177
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