Theorem dminss 2648

Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising."

Assertion

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

Proof of Theorem dminss

Step	Hyp	Ref	Expression
1		19.8a 712	. . . . . . 7 ⊢ ((x ∈ A ∧ xRy) → ∃x(x ∈ A ∧ xRy))
2	1	ancoms 334	. . . . . 6 ⊢ ((xRy ∧ x ∈ A) → ∃x(x ∈ A ∧ xRy))
3		visset 1350	. . . . . . 7 ⊢ y ∈ V
4	3	elima2 2607	. . . . . 6 ⊢ (y ∈ (R “ A) ↔ ∃x(x ∈ A ∧ xRy))
5	2, 4	sylibr 175	. . . . 5 ⊢ ((xRy ∧ x ∈ A) → y ∈ (R “ A))
6		pm3.26 256	. . . . . 6 ⊢ ((xRy ∧ x ∈ A) → xRy)
7		visset 1350	. . . . . . 7 ⊢ x ∈ V
8	3, 7	brcnv 2519	. . . . . 6 ⊢ (y◡Rx ↔ xRy)
9	6, 8	sylibr 175	. . . . 5 ⊢ ((xRy ∧ x ∈ A) → y◡Rx)
10	5, 9	jca 236	. . . 4 ⊢ ((xRy ∧ x ∈ A) → (y ∈ (R “ A) ∧ y◡Rx))
11	10	19.22i 723	. . 3 ⊢ (∃y(xRy ∧ x ∈ A) → ∃y(y ∈ (R “ A) ∧ y◡Rx))
12	7	eldm 2527	. . . . 5 ⊢ (x ∈ dom R ↔ ∃y xRy)
13	12	anbi1i 368	. . . 4 ⊢ ((x ∈ dom R ∧ x ∈ A) ↔ (∃y xRy ∧ x ∈ A))
14		elin 1635	. . . 4 ⊢ (x ∈ (dom R ∩ A) ↔ (x ∈ dom R ∧ x ∈ A))
15		19.41v 963	. . . 4 ⊢ (∃y(xRy ∧ x ∈ A) ↔ (∃y xRy ∧ x ∈ A))
16	13, 14, 15	3bitr4 158	. . 3 ⊢ (x<¶I> ∈ (dom R ∩ A) ↔ ∃y(xRy ∧ x ∈ A))
17	7	elima2 2607	. . 3 ⊢ (x ∈ (◡R “ (R “ A)) ↔ ∃y(y ∈ (R “ A) ∧ y◡Rx))
18	11, 16, 17	3imtr4 192	. 2 ⊢ (x ∈ (dom R ∩ A) → x ∈ (◡R “ (R “ A)))
19	18	ssriv 1508	1 ⊢ (dom R ∩ A) ⊆ (◡R “ (R “ A))

Syntax hints:   ∧ wa 196  ∃wex 678   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487   class class class wbr 2054  ^◡ccnv 2409  dom cdm 2410   “ cima 2413

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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