Theorem dminss 4758

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 1522	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
2	1	ancoms 264	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
3		vex 2604	. . . . . . 7 ⊢ 𝑦 ∈ V
4	3	elima2 4694	. . . . . 6 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
5	2, 4	sylibr 132	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ (𝑅 “ 𝐴))
6		simpl 107	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥𝑅𝑦)
7		vex 2604	. . . . . . 7 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 4536	. . . . . 6 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
9	6, 8	sylibr 132	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦◡𝑅𝑥)
10	5, 9	jca 300	. . . 4 ⊢ ((𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
11	10	eximi 1531	. . 3 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
12	7	eldm 4550	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
13	12	anbi1i 445	. . . 4 ⊢ ((𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
14		elin 3155	. . . 4 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ (𝑥 ∈ dom 𝑅 ∧ 𝑥 ∈ 𝐴))
15		19.41v 1823	. . . 4 ⊢ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
16	13, 14, 15	3bitr4i 210	. . 3 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑥 ∈ 𝐴))
17	7	elima2 4694	. . 3 ⊢ (𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)) ↔ ∃𝑦(𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦◡𝑅𝑥))
18	11, 16, 17	3imtr4i 199	. 2 ⊢ (𝑥 ∈ (dom 𝑅 ∩ 𝐴) → 𝑥 ∈ (◡𝑅 “ (𝑅 “ 𝐴)))
19	18	ssriv 3003	1 ⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴))

Syntax hints:   ∧ wa 102  ∃wex 1421   ∈ wcel 1433   ∩ cin 2972   ⊆ wss 2973   class class class wbr 3785  ^◡ccnv 4362  dom cdm 4363   “ cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)