Theorem dminss 6117

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 2189	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
2	1	ancoms 458	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
3		vex 3433	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elima2 6031	. . . . . 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 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 5837	. . . . . 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 1837	. . 3 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
12	7	eldm 5855	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
13	12	anbi1i 625	. . . 4 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
14		elin 3905	. . . 4 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ (𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴))
15		19.41v 1951	. . . 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 6031	. . 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 3925	1 ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴))

Syntax hints:   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085  ^◡ccnv 5630  dom cdm 5631   “ cima 5634

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-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

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-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  lmhmlsp  21044  cnclsi  23237  kgencn3  23523  kqsat  23696  kqcldsat  23698  cfilucfil  24524  elrgspnsubrunlem2  33309  elrspunidl  33488