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Theorem imagesset 32185
Description: The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
Assertion
Ref Expression
imagesset Image SSet SSet

Proof of Theorem imagesset
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3657 . . . . . . . 8 𝑦𝑦
2 sseq2 3660 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
32rspcev 3340 . . . . . . . 8 ((𝑦𝑥𝑦𝑦) → ∃𝑧𝑥 𝑦𝑧)
41, 3mpan2 707 . . . . . . 7 (𝑦𝑥 → ∃𝑧𝑥 𝑦𝑧)
5 vex 3234 . . . . . . . . 9 𝑦 ∈ V
65elima 5506 . . . . . . . 8 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑧 SSet 𝑦)
7 vex 3234 . . . . . . . . . . 11 𝑧 ∈ V
87, 5brcnv 5337 . . . . . . . . . 10 (𝑧 SSet 𝑦𝑦 SSet 𝑧)
97brsset 32121 . . . . . . . . . 10 (𝑦 SSet 𝑧𝑦𝑧)
108, 9bitri 264 . . . . . . . . 9 (𝑧 SSet 𝑦𝑦𝑧)
1110rexbii 3070 . . . . . . . 8 (∃𝑧𝑥 𝑧 SSet 𝑦 ↔ ∃𝑧𝑥 𝑦𝑧)
126, 11bitri 264 . . . . . . 7 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑦𝑧)
134, 12sylibr 224 . . . . . 6 (𝑦𝑥𝑦 ∈ ( SSet 𝑥))
1413ssriv 3640 . . . . 5 𝑥 ⊆ ( SSet 𝑥)
15 sseq2 3660 . . . . 5 (𝑦 = ( SSet 𝑥) → (𝑥𝑦𝑥 ⊆ ( SSet 𝑥)))
1614, 15mpbiri 248 . . . 4 (𝑦 = ( SSet 𝑥) → 𝑥𝑦)
17 vex 3234 . . . . . 6 𝑥 ∈ V
1817, 5brimage 32158 . . . . 5 (𝑥Image SSet 𝑦𝑦 = ( SSet 𝑥))
19 df-br 4686 . . . . 5 (𝑥Image SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
2018, 19bitr3i 266 . . . 4 (𝑦 = ( SSet 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
215brsset 32121 . . . . 5 (𝑥 SSet 𝑦𝑥𝑦)
22 df-br 4686 . . . . 5 (𝑥 SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2321, 22bitr3i 266 . . . 4 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2416, 20, 233imtr3i 280 . . 3 (⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
2524gen2 1763 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26 funimage 32160 . . 3 Fun Image SSet
27 funrel 5943 . . 3 (Fun Image SSet → Rel Image SSet )
28 ssrel 5241 . . 3 (Rel Image SSet → (Image SSet SSet ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )))
2926, 27, 28mp2b 10 . 2 (Image SSet SSet ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet ))
3025, 29mpbir 221 1 Image SSet SSet
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wcel 2030  wrex 2942  wss 3607  cop 4216   class class class wbr 4685  ccnv 5142  cima 5146  Rel wrel 5148  Fun wfun 5920   SSet csset 32064  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-sset 32088  df-image 32096
This theorem is referenced by: (None)
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