Theorem imagesset 35948

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 3972	. . . . . . . 8 ⊢ 𝑦 ⊆ 𝑦
2		sseq2 3976	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑦))
3	2	rspcev 3591	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ 𝑦) → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
4	1, 3	mpan2 691	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
5		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elima 6039	. . . . . . . 8 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦)
7		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7, 5	brcnv 5849	. . . . . . . . . 10 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 SSet 𝑧)
9	7	brsset 35884	. . . . . . . . . 10 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10	8, 9	bitri 275	. . . . . . . . 9 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 ⊆ 𝑧)
11	10	rexbii 3077	. . . . . . . 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 3953	. . . . 5 ⊢ 𝑥 ⊆ (◡ SSet “ 𝑥)
15		sseq2 3976	. . . . 5 ⊢ (𝑦 = (◡ SSet “ 𝑥) → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ (◡ SSet “ 𝑥)))
16	14, 15	mpbiri 258	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) → 𝑥 ⊆ 𝑦)
17		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
18	17, 5	brimage 35921	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ 𝑦 = (◡ SSet “ 𝑥))
19		df-br 5111	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
20	18, 19	bitr3i 277	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
21	5	brsset 35884	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ 𝑥 ⊆ 𝑦)
22		df-br 5111	. . . . 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 35923	. . 3 ⊢ Fun Image◡ SSet
27		funrel 6536	. . 3 ⊢ (Fun Image◡ SSet → Rel Image◡ SSet )
28		ssrel 5748	. . 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 3054   ⊆ wss 3917  ⟨cop 4598   class class class wbr 5110  ^◡ccnv 5640   “ cima 5644  Rel wrel 5646  Fun wfun 6508   SSet csset 35827  Imagecimage 35835

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-symdif 4219  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-1st 7971  df-2nd 7972  df-txp 35849  df-sset 35851  df-image 35859

This theorem is referenced by: (None)