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Theorem imagesset 34994
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 4004 . . . . . . . 8 𝑦𝑦
2 sseq2 4008 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
32rspcev 3612 . . . . . . . 8 ((𝑦𝑥𝑦𝑦) → ∃𝑧𝑥 𝑦𝑧)
41, 3mpan2 689 . . . . . . 7 (𝑦𝑥 → ∃𝑧𝑥 𝑦𝑧)
5 vex 3478 . . . . . . . . 9 𝑦 ∈ V
65elima 6064 . . . . . . . 8 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑧 SSet 𝑦)
7 vex 3478 . . . . . . . . . . 11 𝑧 ∈ V
87, 5brcnv 5882 . . . . . . . . . 10 (𝑧 SSet 𝑦𝑦 SSet 𝑧)
97brsset 34930 . . . . . . . . . 10 (𝑦 SSet 𝑧𝑦𝑧)
108, 9bitri 274 . . . . . . . . 9 (𝑧 SSet 𝑦𝑦𝑧)
1110rexbii 3094 . . . . . . . 8 (∃𝑧𝑥 𝑧 SSet 𝑦 ↔ ∃𝑧𝑥 𝑦𝑧)
126, 11bitri 274 . . . . . . 7 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑦𝑧)
134, 12sylibr 233 . . . . . 6 (𝑦𝑥𝑦 ∈ ( SSet 𝑥))
1413ssriv 3986 . . . . 5 𝑥 ⊆ ( SSet 𝑥)
15 sseq2 4008 . . . . 5 (𝑦 = ( SSet 𝑥) → (𝑥𝑦𝑥 ⊆ ( SSet 𝑥)))
1614, 15mpbiri 257 . . . 4 (𝑦 = ( SSet 𝑥) → 𝑥𝑦)
17 vex 3478 . . . . . 6 𝑥 ∈ V
1817, 5brimage 34967 . . . . 5 (𝑥Image SSet 𝑦𝑦 = ( SSet 𝑥))
19 df-br 5149 . . . . 5 (𝑥Image SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
2018, 19bitr3i 276 . . . 4 (𝑦 = ( SSet 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
215brsset 34930 . . . . 5 (𝑥 SSet 𝑦𝑥𝑦)
22 df-br 5149 . . . . 5 (𝑥 SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2321, 22bitr3i 276 . . . 4 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
2416, 20, 233imtr3i 290 . . 3 (⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
2524gen2 1798 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26 funimage 34969 . . 3 Fun Image SSet
27 funrel 6565 . . 3 (Fun Image SSet → Rel Image SSet )
28 ssrel 5782 . . 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 230 1 Image SSet SSet
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  wrex 3070  wss 3948  cop 4634   class class class wbr 5148  ccnv 5675  cima 5679  Rel wrel 5681  Fun wfun 6537   SSet csset 34873  Imagecimage 34881
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977  df-2nd 7978  df-txp 34895  df-sset 34897  df-image 34905
This theorem is referenced by: (None)
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