Theorem imagesset 36151

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 3945	. . . . . . . 8 ⊢ 𝑦 ⊆ 𝑦
2		sseq2 3949	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ 𝑦))
3	2	rspcev 3565	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ 𝑦) → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
4	1, 3	mpan2 692	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑥 𝑦 ⊆ 𝑧)
5		vex 3434	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	5	elima 6024	. . . . . . . 8 ⊢ (𝑦 ∈ (◡ SSet “ 𝑥) ↔ ∃𝑧 ∈ 𝑥 𝑧◡ SSet 𝑦)
7		vex 3434	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7, 5	brcnv 5831	. . . . . . . . . 10 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 SSet 𝑧)
9	7	brsset 36085	. . . . . . . . . 10 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10	8, 9	bitri 275	. . . . . . . . 9 ⊢ (𝑧◡ SSet 𝑦 ↔ 𝑦 ⊆ 𝑧)
11	10	rexbii 3085	. . . . . . . 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 3926	. . . . 5 ⊢ 𝑥 ⊆ (◡ SSet “ 𝑥)
15		sseq2 3949	. . . . 5 ⊢ (𝑦 = (◡ SSet “ 𝑥) → (𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ (◡ SSet “ 𝑥)))
16	14, 15	mpbiri 258	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) → 𝑥 ⊆ 𝑦)
17		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
18	17, 5	brimage 36122	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ 𝑦 = (◡ SSet “ 𝑥))
19		df-br 5087	. . . . 5 ⊢ (𝑥Image◡ SSet 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
20	18, 19	bitr3i 277	. . . 4 ⊢ (𝑦 = (◡ SSet “ 𝑥) ↔ ⟨𝑥, 𝑦⟩ ∈ Image◡ SSet )
21	5	brsset 36085	. . . . 5 ⊢ (𝑥 SSet 𝑦 ↔ 𝑥 ⊆ 𝑦)
22		df-br 5087	. . . . 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 1798	. 2 ⊢ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ Image◡ SSet → ⟨𝑥, 𝑦⟩ ∈ SSet )
26		funimage 36124	. . 3 ⊢ Fun Image◡ SSet
27		funrel 6509	. . 3 ⊢ (Fun Image◡ SSet → Rel Image◡ SSet )
28		ssrel 5732	. . 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 1540   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086  ^◡ccnv 5623   “ cima 5627  Rel wrel 5629  Fun wfun 6486   SSet csset 36028  Imagecimage 36036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7935  df-2nd 7936  df-txp 36050  df-sset 36052  df-image 36060

This theorem is referenced by: (None)