Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  imagesset Structured version   Visualization version   GIF version

Theorem imagesset 31699
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 3603 . . . . . . . 8 𝑦𝑦
2 sseq2 3606 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
32rspcev 3295 . . . . . . . 8 ((𝑦𝑥𝑦𝑦) → ∃𝑧𝑥 𝑦𝑧)
41, 3mpan2 706 . . . . . . 7 (𝑦𝑥 → ∃𝑧𝑥 𝑦𝑧)
5 vex 3189 . . . . . . . . 9 𝑦 ∈ V
65elima 5430 . . . . . . . 8 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑧 SSet 𝑦)
7 vex 3189 . . . . . . . . . . 11 𝑧 ∈ V
87, 5brcnv 5265 . . . . . . . . . 10 (𝑧 SSet 𝑦𝑦 SSet 𝑧)
97brsset 31635 . . . . . . . . . 10 (𝑦 SSet 𝑧𝑦𝑧)
108, 9bitri 264 . . . . . . . . 9 (𝑧 SSet 𝑦𝑦𝑧)
1110rexbii 3034 . . . . . . . 8 (∃𝑧𝑥 𝑧 SSet 𝑦 ↔ ∃𝑧𝑥 𝑦𝑧)
126, 11bitri 264 . . . . . . 7 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑦𝑧)
134, 12sylibr 224 . . . . . 6 (𝑦𝑥𝑦 ∈ ( SSet 𝑥))
1413ssriv 3587 . . . . 5 𝑥 ⊆ ( SSet 𝑥)
15 sseq2 3606 . . . . 5 (𝑦 = ( SSet 𝑥) → (𝑥𝑦𝑥 ⊆ ( SSet 𝑥)))
1614, 15mpbiri 248 . . . 4 (𝑦 = ( SSet 𝑥) → 𝑥𝑦)
17 vex 3189 . . . . . 6 𝑥 ∈ V
1817, 5brimage 31672 . . . . 5 (𝑥Image SSet 𝑦𝑦 = ( SSet 𝑥))
19 df-br 4614 . . . . 5 (𝑥Image SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
2018, 19bitr3i 266 . . . 4 (𝑦 = ( SSet 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
215brsset 31635 . . . . 5 (𝑥 SSet 𝑦𝑥𝑦)
22 df-br 4614 . . . . 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 1720 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26 funimage 31674 . . 3 Fun Image SSet
27 funrel 5864 . . 3 (Fun Image SSet → Rel Image SSet )
28 ssrel 5168 . . 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 1478   = wceq 1480  wcel 1987  wrex 2908  wss 3555  cop 4154   class class class wbr 4613  ccnv 5073  cima 5077  Rel wrel 5079  Fun wfun 5841   SSet csset 31577  Imagecimage 31585
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-symdif 3822  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31599  df-sset 31601  df-image 31609
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator