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Theorem imagesset 32358
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 3757 . . . . . . . 8 𝑦𝑦
2 sseq2 3760 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
32rspcev 3441 . . . . . . . 8 ((𝑦𝑥𝑦𝑦) → ∃𝑧𝑥 𝑦𝑧)
41, 3mpan2 709 . . . . . . 7 (𝑦𝑥 → ∃𝑧𝑥 𝑦𝑧)
5 vex 3335 . . . . . . . . 9 𝑦 ∈ V
65elima 5621 . . . . . . . 8 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑧 SSet 𝑦)
7 vex 3335 . . . . . . . . . . 11 𝑧 ∈ V
87, 5brcnv 5452 . . . . . . . . . 10 (𝑧 SSet 𝑦𝑦 SSet 𝑧)
97brsset 32294 . . . . . . . . . 10 (𝑦 SSet 𝑧𝑦𝑧)
108, 9bitri 264 . . . . . . . . 9 (𝑧 SSet 𝑦𝑦𝑧)
1110rexbii 3171 . . . . . . . 8 (∃𝑧𝑥 𝑧 SSet 𝑦 ↔ ∃𝑧𝑥 𝑦𝑧)
126, 11bitri 264 . . . . . . 7 (𝑦 ∈ ( SSet 𝑥) ↔ ∃𝑧𝑥 𝑦𝑧)
134, 12sylibr 224 . . . . . 6 (𝑦𝑥𝑦 ∈ ( SSet 𝑥))
1413ssriv 3740 . . . . 5 𝑥 ⊆ ( SSet 𝑥)
15 sseq2 3760 . . . . 5 (𝑦 = ( SSet 𝑥) → (𝑥𝑦𝑥 ⊆ ( SSet 𝑥)))
1614, 15mpbiri 248 . . . 4 (𝑦 = ( SSet 𝑥) → 𝑥𝑦)
17 vex 3335 . . . . . 6 𝑥 ∈ V
1817, 5brimage 32331 . . . . 5 (𝑥Image SSet 𝑦𝑦 = ( SSet 𝑥))
19 df-br 4797 . . . . 5 (𝑥Image SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
2018, 19bitr3i 266 . . . 4 (𝑦 = ( SSet 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image SSet )
215brsset 32294 . . . . 5 (𝑥 SSet 𝑦𝑥𝑦)
22 df-br 4797 . . . . 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 1864 . 2 𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ Image SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26 funimage 32333 . . 3 Fun Image SSet
27 funrel 6058 . . 3 (Fun Image SSet → Rel Image SSet )
28 ssrel 5356 . . 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 1622   = wceq 1624  wcel 2131  wrex 3043  wss 3707  cop 4319   class class class wbr 4796  ccnv 5257  cima 5261  Rel wrel 5263  Fun wfun 6035   SSet csset 32237  Imagecimage 32245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-symdif 3979  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-eprel 5171  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-1st 7325  df-2nd 7326  df-txp 32259  df-sset 32261  df-image 32269
This theorem is referenced by: (None)
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