Theorem clos1eq2 5876

Description: Equality law for closure. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
clos1eq2	|- (R = T -> Clos1 (S, R) = Clos1 (S, T))

Proof of Theorem clos1eq2

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imaeq1 4938	. . . . . 6 |- (R = T -> (R"a) = (T"a))
2	1	sseq1d 3299	. . . . 5 |- (R = T -> ((R"a) C_ a <-> (T"a) C_ a))
3	2	anbi2d 684	. . . 4 |- (R = T -> ((S C_ a /\ (R"a) C_ a) <-> (S C_ a /\ (T"a) C_ a)))
4	3	abbidv 2468	. . 3 |- (R = T -> {a | (S C_ a /\ (R"a) C_ a)} = {a | (S C_ a /\ (T"a) C_ a)})
5		inteq 3930	. . 3 |- ({a | (S C_ a /\ (R"a) C_ a)} = {a | (S C_ a /\ (T"a) C_ a)} -> |^|{a | (S C_ a /\ (R"a) C_ a)} = |^|{a | (S C_ a /\ (T"a) C_ a)})
6	4, 5	syl 15	. 2 |- (R = T -> |^|{a | (S C_ a /\ (R"a) C_ a)} = |^|{a | (S C_ a /\ (T"a) C_ a)})
7		df-clos1 5874	. 2 |- Clos1 (S, R) = |^|{a | (S C_ a /\ (R"a) C_ a)}
8		df-clos1 5874	. 2 |- Clos1 (S, T) = |^|{a | (S C_ a /\ (T"a) C_ a)}
9	6, 7, 8	3eqtr4g 2410	1 |- (R = T -> Clos1 (S, R) = Clos1 (S, T))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642  {cab 2339   C_ wss 3258  |^|cint 3927  "cima 4723   Clos1 cclos1 5873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928  df-br 4641  df-ima 4728  df-clos1 5874

This theorem is referenced by:  clos1exg  5878  clos1basesucg  5885  freceq12  6312