Theorem imagesset 36316

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 3961	. . . . . . . 8 ⊢ 𝑦 ⊆ 𝑦
2		sseq2 3965	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑦))
3	2	rspcev 3584	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ 𝑦) → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
4	1, 3	mpan2 703	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
5		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elima 6058	. . . . . . . 8 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦)
7		vex 3461	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7, 5	brcnv 5859	. . . . . . . . . 10 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 SSet 𝑧)
9	7	brsset 36250	. . . . . . . . . 10 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10	8, 9	bitri 278	. . . . . . . . 9 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 ⊆ 𝑧)
11	10	rexbii 3112	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
12	6, 11	bitri 278	. . . . . . 7 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
13	4, 12	sylibr 237	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ∈ (◡ SSet “ 𝑥))
14	13	ssriv 3943	. . . . 5 ⊢ 𝑥 ⊆ (◡ SSet “ 𝑥)
15		sseq2 3965	. . . . 5 ⊢ (𝑦 = (◡ SSet “ 𝑥) → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ (◡ SSet “ 𝑥)))
16	14, 15	mpbiri 261	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) → 𝑥 ⊆ 𝑦)
17		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
18	17, 5	brimage 36287	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ 𝑦 = (◡ SSet “ 𝑥))
19		df-br 5106	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
20	18, 19	bitr3i 280	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
21	5	brsset 36250	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ 𝑥 ⊆ 𝑦)
22		df-br 5106	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
23	21, 22	bitr3i 280	. . . 4 ⊢ (𝑥 ⊆ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ SSet )
24	16, 20, 23	3imtr3i 294	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
25	24	gen2 1819	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26		funimage 36289	. . 3 ⊢ Fun Image◡ SSet
27		funrel 6542	. . 3 ⊢ (Fun Image◡ SSet → Rel Image◡ SSet )
28		ssrel 5760	. . 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 234	1 ⊢ Image◡ SSet ⊆ SSet

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   ⊆ wss 3907  ⟨cop 4591   class class class wbr 5105  ^◡ccnv 5651   “ cima 5655  Rel wrel 5657  Fun wfun 6519   SSet csset 36193  Imagecimage 36201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-symdif 4208  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-1st 7974  df-2nd 7975  df-txp 36215  df-sset 36217  df-image 36225

This theorem is referenced by: (None)