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Theorem funpartlem 31744
Description: Lemma for funpartfun 31745. 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 3202 . 2 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2 vsnid 4187 . . . . 5 𝑥 ∈ {𝑥}
3 eleq2 2687 . . . . 5 ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
42, 3mpbiri 248 . . . 4 ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5 n0i 3902 . . . . 5 (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6 snprc 4230 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 206 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
87imaeq2d 5435 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9 ima0 5450 . . . . . 6 (𝐹 “ ∅) = ∅
108, 9syl6eq 2671 . . . . 5 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
115, 10nsyl2 142 . . . 4 (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
124, 11syl 17 . . 3 ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
1312exlimiv 1855 . 2 (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14 eleq1 2686 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15 sneq 4165 . . . . . 6 (𝑦 = 𝐴 → {𝑦} = {𝐴})
1615imaeq2d 5435 . . . . 5 (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
1716eqeq1d 2623 . . . 4 (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
1817exbidv 1847 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19 vex 3193 . . . . 5 𝑦 ∈ V
2019eldm 5291 . . . 4 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21 brxp 5117 . . . . . . . . . 10 (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 Singletons ))
2219, 21mpbiran 952 . . . . . . . . 9 (𝑦(V × Singletons )𝑧𝑧 Singletons )
23 elsingles 31720 . . . . . . . . 9 (𝑧 Singletons ↔ ∃𝑥 𝑧 = {𝑥})
2422, 23bitri 264 . . . . . . . 8 (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
2524anbi2i 729 . . . . . . 7 ((𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26 brin 4674 . . . . . . 7 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧))
27 19.42v 1915 . . . . . . 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 1771 . . . . 5 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
30 excom 2039 . . . . 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 1784 . . . . . 6 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33 snex 4879 . . . . . . 7 {𝑥} ∈ V
34 breq2 4627 . . . . . . 7 (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(Image𝐹 ∘ Singleton){𝑥}))
3533, 34ceqsexv 3232 . . . . . 6 (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
3619, 33brco 5262 . . . . . . 7 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}))
37 vex 3193 . . . . . . . . . 10 𝑧 ∈ V
3819, 37brsingle 31719 . . . . . . . . 9 (𝑦Singleton𝑧𝑧 = {𝑦})
3938anbi1i 730 . . . . . . . 8 ((𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
4039exbii 1771 . . . . . . 7 (∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41 snex 4879 . . . . . . . . 9 {𝑦} ∈ V
42 breq1 4626 . . . . . . . . 9 (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
4341, 42ceqsexv 3232 . . . . . . . 8 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
4441, 33brimage 31728 . . . . . . . 8 ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45 eqcom 2628 . . . . . . . 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 1771 . . . 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 3257 . 2 (𝐴 ∈ V → (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
521, 13, 51pm5.21nii 368 1 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  cin 3559  c0 3897  {csn 4155   class class class wbr 4623   × cxp 5082  dom cdm 5084  cima 5087  ccom 5088  Singletoncsingle 31639   Singletons csingles 31640  Imagecimage 31641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-symdif 3828  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-eprel 4995  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-singleton 31663  df-singles 31664  df-image 31665
This theorem is referenced by:  funpartfun  31745  funpartfv  31747
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