Theorem dminss 6106

Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
dminss	⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴))

Proof of Theorem dminss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 2182	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
2	1	ancoms 458	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
3		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elima2 6021	. . . . . 6 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
5	2, 4	sylibr 234	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ (𝑅 “ 𝐴))
6		simpl 482	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥𝑅𝑦)
7		vex 3442	. . . . . . 7 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 5829	. . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	6, 8	sylibr 234	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦◡𝑅𝑥)
10	5, 9	jca 511	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
11	10	eximi 1835	. . 3 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
12	7	eldm 5847	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
13	12	anbi1i 624	. . . 4 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
14		elin 3921	. . . 4 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ (𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴))
15		19.41v 1949	. . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
16	13, 14, 15	3bitr4i 303	. . 3 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
17	7	elima2 6021	. . 3 ⊢ (𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
18	11, 16, 17	3imtr4i 292	. 2 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)))
19	18	ssriv 3941	1 ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴))

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5095  ^◡ccnv 5622  dom cdm 5623   “ cima 5626

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-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  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-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  lmhmlsp  20971  cnclsi  23175  kgencn3  23461  kqsat  23634  kqcldsat  23636  cfilucfil  24463  elrgspnsubrunlem2  33201  elrspunidl  33378