Theorem imagesset 36540

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 3956	. . . . . . . 8 ⊢ 𝑦 ⊆ 𝑦
2		sseq2 3960	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑦))
3	2	rspcev 3579	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ 𝑦) → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
4	1, 3	mpan2 704	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
5		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elima 6065	. . . . . . . 8 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦)
7		vex 3457	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7, 5	brcnv 5866	. . . . . . . . . 10 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 SSet 𝑧)
9	7	brsset 36474	. . . . . . . . . 10 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10	8, 9	bitri 278	. . . . . . . . 9 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 ⊆ 𝑧)
11	10	rexbii 3111	. . . . . . . 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 3938	. . . . 5 ⊢ 𝑥 ⊆ (◡ SSet “ 𝑥)
15		sseq2 3960	. . . . 5 ⊢ (𝑦 = (◡ SSet “ 𝑥) → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ (◡ SSet “ 𝑥)))
16	14, 15	mpbiri 261	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) → 𝑥 ⊆ 𝑦)
17		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
18	17, 5	brimage 36511	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ 𝑦 = (◡ SSet “ 𝑥))
19		df-br 5108	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
20	18, 19	bitr3i 280	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
21	5	brsset 36474	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ 𝑥 ⊆ 𝑦)
22		df-br 5108	. . . . 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 1829	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26		funimage 36513	. . 3 ⊢ Fun Image◡ SSet
27		funrel 6554	. . 3 ⊢ (Fun Image◡ SSet → Rel Image◡ SSet )
28		ssrel 5767	. . 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

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  ⟨cop 4593   class class class wbr 5107  ^◡ccnv 5658   “ cima 5662  Rel wrel 5664  Fun wfun 6531   SSet csset 36417  Imagecimage 36425

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4202  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7990  df-2nd 7991  df-txp 36439  df-sset 36441  df-image 36449

This theorem is used by: (None)