Theorem dminss 4681

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 1479	. . . . . . 7 ⊢ ((x ∈ A ∧ x𝑅y) → ∃x(x ∈ A ∧ x𝑅y))
2	1	ancoms 255	. . . . . 6 ⊢ ((x𝑅y ∧ x ∈ A) → ∃x(x ∈ A ∧ x𝑅y))
3		vex 2554	. . . . . . 7 ⊢ y ∈ V
4	3	elima2 4617	. . . . . 6 ⊢ (y ∈ (𝑅 “ A) ↔ ∃x(x ∈ A ∧ x𝑅y))
5	2, 4	sylibr 137	. . . . 5 ⊢ ((x𝑅y ∧ x ∈ A) → y ∈ (𝑅 “ A))
6		simpl 102	. . . . . 6 ⊢ ((x𝑅y ∧ x ∈ A) → x𝑅y)
7		vex 2554	. . . . . . 7 ⊢ x ∈ V
8	3, 7	brcnv 4461	. . . . . 6 ⊢ (y◡𝑅x ↔ x𝑅y)
9	6, 8	sylibr 137	. . . . 5 ⊢ ((x𝑅y ∧ x ∈ A) → y◡𝑅x)
10	5, 9	jca 290	. . . 4 ⊢ ((x𝑅y ∧ x ∈ A) → (y ∈ (𝑅 “ A) ∧ y◡𝑅x))
11	10	eximi 1488	. . 3 ⊢ (∃y(x𝑅y ∧ x ∈ A) → ∃y(y ∈ (𝑅 “ A) ∧ y◡𝑅x))
12	7	eldm 4475	. . . . 5 ⊢ (x ∈ dom 𝑅 ↔ ∃y x𝑅y)
13	12	anbi1i 431	. . . 4 ⊢ ((x ∈ dom 𝑅 ∧ x ∈ A) ↔ (∃y x𝑅y ∧ x ∈ A))
14		elin 3120	. . . 4 ⊢ (x ∈ (dom 𝑅 ∩ A) ↔ (x ∈ dom 𝑅 ∧ x ∈ A))
15		19.41v 1779	. . . 4 ⊢ (∃y(x𝑅y ∧ x ∈ A) ↔ (∃y x𝑅y ∧ x ∈ A))
16	13, 14, 15	3bitr4i 201	. . 3 ⊢ (x ∈ (dom 𝑅 ∩ A) ↔ ∃y(x𝑅y ∧ x ∈ A))
17	7	elima2 4617	. . 3 ⊢ (x ∈ (◡𝑅 “ (𝑅 “ A)) ↔ ∃y(y ∈ (𝑅 “ A) ∧ y◡𝑅x))
18	11, 16, 17	3imtr4i 190	. 2 ⊢ (x ∈ (dom 𝑅 ∩ A) → x ∈ (◡𝑅 “ (𝑅 “ A)))
19	18	ssriv 2943	1 ⊢ (dom 𝑅 ∩ A) ⊆ (◡𝑅 “ (𝑅 “ A))

Syntax hints:   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911   class class class wbr 3755  ^◡ccnv 4287  dom cdm 4288   “ cima 4291

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by: (None)