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Theorem imagesset 33471
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 3975 . . . . . . . 8 𝑦𝑦
2 sseq2 3979 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
32rspcev 3609 . . . . . . . 8 ((𝑦𝑥𝑦𝑦) → ∃𝑧𝑥 𝑦𝑧)
41, 3mpan2 690 . . . . . . 7 (𝑦𝑥 → ∃𝑧𝑥 𝑦𝑧)
5 vex 3483 . . . . . . . . 9 𝑦 ∈ V
65elima 5921 . . . . . . . 8 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑧 SSet 𝑦)
7 vex 3483 . . . . . . . . . . 11 𝑧 ∈ V
87, 5brcnv 5740 . . . . . . . . . 10 (𝑧 SSet 𝑦𝑦 SSet 𝑧)
97brsset 33407 . . . . . . . . . 10 (𝑦 SSet 𝑧𝑦𝑧)
108, 9bitri 278 . . . . . . . . 9 (𝑧 SSet 𝑦𝑦𝑧)
1110rexbii 3241 . . . . . . . 8 (∃𝑧𝑥 𝑧 SSet 𝑦 ↔ ∃𝑧𝑥 𝑦𝑧)
126, 11bitri 278 . . . . . . 7 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑦𝑧)
134, 12sylibr 237 . . . . . 6 (𝑦𝑥𝑦 ∈ ( SSet 𝑥))
1413ssriv 3957 . . . . 5 𝑥 ⊆ ( SSet 𝑥)
15 sseq2 3979 . . . . 5 (𝑦 = ( SSet 𝑥) → (𝑥𝑦𝑥 ⊆ ( SSet 𝑥)))
1614, 15mpbiri 261 . . . 4 (𝑦 = ( SSet 𝑥) → 𝑥𝑦)
17 vex 3483 . . . . . 6 𝑥 ∈ V
1817, 5brimage 33444 . . . . 5 (𝑥Image SSet 𝑦𝑦 = ( SSet 𝑥))
19 df-br 5053 . . . . 5 (𝑥Image SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
2018, 19bitr3i 280 . . . 4 (𝑦 = ( SSet 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
215brsset 33407 . . . . 5 (𝑥 SSet 𝑦𝑥𝑦)
22 df-br 5053 . . . . 5 (𝑥 SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2321, 22bitr3i 280 . . . 4 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2416, 20, 233imtr3i 294 . . 3 (⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
2524gen2 1798 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26 funimage 33446 . . 3 Fun Image SSet
27 funrel 6360 . . 3 (Fun Image SSet → Rel Image SSet )
28 ssrel 5644 . . 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 234 1 Image SSet SSet
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wcel 2115  wrex 3134  wss 3919  cop 4556   class class class wbr 5052  ccnv 5541  cima 5545  Rel wrel 5547  Fun wfun 6337   SSet csset 33350  Imagecimage 33358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-symdif 4204  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-eprel 5452  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351  df-1st 7684  df-2nd 7685  df-txp 33372  df-sset 33374  df-image 33382
This theorem is referenced by: (None)
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