Theorem imagesset 35926

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.

Step	Hyp	Ref	Expression
1		ssid 3960	. . . . . . . 8 ⊢ 𝑦 ⊆ 𝑦
2		sseq2 3964	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑦))
3	2	rspcev 3579	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ 𝑦) → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
4	1, 3	mpan2 691	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
5		vex 3442	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elima 6020	. . . . . . . 8 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦)
7		vex 3442	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7, 5	brcnv 5829	. . . . . . . . . 10 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 SSet 𝑧)
9	7	brsset 35862	. . . . . . . . . 10 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10	8, 9	bitri 275	. . . . . . . . 9 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 ⊆ 𝑧)
11	10	rexbii 3076	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
12	6, 11	bitri 275	. . . . . . 7 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
13	4, 12	sylibr 234	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ∈ (◡ SSet “ 𝑥))
14	13	ssriv 3941	. . . . 5 ⊢ 𝑥 ⊆ (◡ SSet “ 𝑥)
15		sseq2 3964	. . . . 5 ⊢ (𝑦 = (◡ SSet “ 𝑥) → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ (◡ SSet “ 𝑥)))
16	14, 15	mpbiri 258	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) → 𝑥 ⊆ 𝑦)
17		vex 3442	. . . . . 6 ⊢ 𝑥 ∈ V
18	17, 5	brimage 35899	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ 𝑦 = (◡ SSet “ 𝑥))
19		df-br 5096	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
20	18, 19	bitr3i 277	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
21	5	brsset 35862	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ 𝑥 ⊆ 𝑦)
22		df-br 5096	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
23	21, 22	bitr3i 277	. . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
24	16, 20, 23	3imtr3i 291	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
25	24	gen2 1796	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26		funimage 35901	. . 3 ⊢ Fun Image◡ SSet
27		funrel 6503	. . 3 ⊢ (Fun Image◡ SSet → Rel Image◡ SSet )
28		ssrel 5730	. . 3 ⊢ (Rel Image◡ SSet → (Image◡ SSet ⊆ SSet ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )))
29	26, 27, 28	mp2b 10	. 2 ⊢ (Image◡ SSet ⊆ SSet ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet ))
30	25, 29	mpbir 231	1 ⊢ Image◡ SSet ⊆ SSet

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3905  ⟨cop 4585   class class class wbr 5095  ^◡ccnv 5622   “ cima 5626  Rel wrel 5628  Fun wfun 6480   SSet csset 35805  Imagecimage 35813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fo 6492  df-fv 6494  df-1st 7931  df-2nd 7932  df-txp 35827  df-sset 35829  df-image 35837

This theorem is referenced by: (None)