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Theorem funpartlem 31219
 Description: Lemma for funpartfun 31220. 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 3185 . 2 (𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) → 𝐴 ∈ V)
2 vsnid 4156 . . . . 5 𝑥 ∈ {𝑥}
3 eleq2 2677 . . . . 5 ((𝐹 “ {𝐴}) = {𝑥} → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ {𝑥}))
42, 3mpbiri 247 . . . 4 ((𝐹 “ {𝐴}) = {𝑥} → 𝑥 ∈ (𝐹 “ {𝐴}))
5 n0i 3879 . . . . 5 (𝑥 ∈ (𝐹 “ {𝐴}) → ¬ (𝐹 “ {𝐴}) = ∅)
6 snprc 4197 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
76biimpi 205 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
87imaeq2d 5385 . . . . . 6 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
9 ima0 5400 . . . . . 6 (𝐹 “ ∅) = ∅
108, 9syl6eq 2660 . . . . 5 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
115, 10nsyl2 141 . . . 4 (𝑥 ∈ (𝐹 “ {𝐴}) → 𝐴 ∈ V)
124, 11syl 17 . . 3 ((𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
1312exlimiv 1845 . 2 (∃𝑥(𝐹 “ {𝐴}) = {𝑥} → 𝐴 ∈ V)
14 eleq1 2676 . . 3 (𝑦 = 𝐴 → (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ 𝐴 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))))
15 sneq 4135 . . . . . 6 (𝑦 = 𝐴 → {𝑦} = {𝐴})
1615imaeq2d 5385 . . . . 5 (𝑦 = 𝐴 → (𝐹 “ {𝑦}) = (𝐹 “ {𝐴}))
1716eqeq1d 2612 . . . 4 (𝑦 = 𝐴 → ((𝐹 “ {𝑦}) = {𝑥} ↔ (𝐹 “ {𝐴}) = {𝑥}))
1817exbidv 1837 . . 3 (𝑦 = 𝐴 → (∃𝑥(𝐹 “ {𝑦}) = {𝑥} ↔ ∃𝑥(𝐹 “ {𝐴}) = {𝑥}))
19 vex 3176 . . . . 5 𝑦 ∈ V
2019eldm 5243 . . . 4 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧)
21 brxp 5071 . . . . . . . . . 10 (𝑦(V × Singletons )𝑧 ↔ (𝑦 ∈ V ∧ 𝑧 Singletons ))
2219, 21mpbiran 955 . . . . . . . . 9 (𝑦(V × Singletons )𝑧𝑧 Singletons )
23 elsingles 31195 . . . . . . . . 9 (𝑧 Singletons ↔ ∃𝑥 𝑧 = {𝑥})
2422, 23bitri 263 . . . . . . . 8 (𝑦(V × Singletons )𝑧 ↔ ∃𝑥 𝑧 = {𝑥})
2524anbi2i 726 . . . . . . 7 ((𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
26 brin 4634 . . . . . . 7 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(V × Singletons )𝑧))
27 19.42v 1905 . . . . . . 7 (∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝑦(Image𝐹 ∘ Singleton)𝑧 ∧ ∃𝑥 𝑧 = {𝑥}))
2825, 26, 273bitr4i 291 . . . . . 6 (𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
2928exbii 1764 . . . . 5 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
30 excom 2029 . . . . 5 (∃𝑧𝑥(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
3129, 30bitri 263 . . . 4 (∃𝑧 𝑦((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))𝑧 ↔ ∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}))
32 exancom 1774 . . . . . 6 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧))
33 snex 4835 . . . . . . 7 {𝑥} ∈ V
34 breq2 4587 . . . . . . 7 (𝑧 = {𝑥} → (𝑦(Image𝐹 ∘ Singleton)𝑧𝑦(Image𝐹 ∘ Singleton){𝑥}))
3533, 34ceqsexv 3215 . . . . . 6 (∃𝑧(𝑧 = {𝑥} ∧ 𝑦(Image𝐹 ∘ Singleton)𝑧) ↔ 𝑦(Image𝐹 ∘ Singleton){𝑥})
3619, 33brco 5214 . . . . . . 7 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ ∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}))
37 vex 3176 . . . . . . . . . 10 𝑧 ∈ V
3819, 37brsingle 31194 . . . . . . . . 9 (𝑦Singleton𝑧𝑧 = {𝑦})
3938anbi1i 727 . . . . . . . 8 ((𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ (𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
4039exbii 1764 . . . . . . 7 (∃𝑧(𝑦Singleton𝑧𝑧Image𝐹{𝑥}) ↔ ∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}))
41 snex 4835 . . . . . . . . 9 {𝑦} ∈ V
42 breq1 4586 . . . . . . . . 9 (𝑧 = {𝑦} → (𝑧Image𝐹{𝑥} ↔ {𝑦}Image𝐹{𝑥}))
4341, 42ceqsexv 3215 . . . . . . . 8 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ {𝑦}Image𝐹{𝑥})
4441, 33brimage 31203 . . . . . . . 8 ({𝑦}Image𝐹{𝑥} ↔ {𝑥} = (𝐹 “ {𝑦}))
45 eqcom 2617 . . . . . . . 8 ({𝑥} = (𝐹 “ {𝑦}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4643, 44, 453bitri 285 . . . . . . 7 (∃𝑧(𝑧 = {𝑦} ∧ 𝑧Image𝐹{𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4736, 40, 463bitri 285 . . . . . 6 (𝑦(Image𝐹 ∘ Singleton){𝑥} ↔ (𝐹 “ {𝑦}) = {𝑥})
4832, 35, 473bitri 285 . . . . 5 (∃𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ (𝐹 “ {𝑦}) = {𝑥})
4948exbii 1764 . . . 4 (∃𝑥𝑧(𝑦(Image𝐹 ∘ Singleton)𝑧𝑧 = {𝑥}) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5020, 31, 493bitri 285 . . 3 (𝑦 ∈ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )) ↔ ∃𝑥(𝐹 “ {𝑦}) = {𝑥})
5114, 18, 50vtoclbg 3240 . 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  dom cdm 5038   “ cima 5041   ∘ ccom 5042  Singletoncsingle 31114   Singletons csingles 31115  Imagecimage 31116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139  df-image 31140 This theorem is referenced by:  funpartfun  31220  funpartfv  31222
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