Theorem ivth

Description: The intermediate value theorem, increasing case. This is Metamath 100 proof #79. (Contributed by Paul Chapman, 22-Jan-2008) (Proof shortened by Mario Carneiro, 30-Apr-2014)

	Ref	Expression
Hypotheses	ivth.1	|- ( ph -> A e. RR )
	ivth.2	|- ( ph -> B e. RR )
	ivth.3	|- ( ph -> U e. RR )
	ivth.4	|- ( ph -> A < B )
	ivth.5	|- ( ph -> ( A [,] B ) C_ D )
	ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
	ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
	ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
Assertion	ivth	|- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )

Proof

Step	Hyp	Ref	Expression
1		ivth.1	|- ( ph -> A e. RR )
2		ivth.2	|- ( ph -> B e. RR )
3		ivth.3	|- ( ph -> U e. RR )
4		ivth.4	|- ( ph -> A < B )
5		ivth.5	|- ( ph -> ( A [,] B ) C_ D )
6		ivth.7	|- ( ph -> F e. ( D -cn-> CC ) )
7		ivth.8	|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )
8		ivth.9	|- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )
9		fveq2	|- ( y = x -> ( F ` y ) = ( F ` x ) )
10	9	breq1d	|- ( y = x -> ( ( F ` y ) <_ U <-> ( F ` x ) <_ U ) )
11	10	cbvrabv	|- { y e. ( A [,] B ) | ( F ` y ) <_ U } = { x e. ( A [,] B ) | ( F ` x ) <_ U }
12		eqid	|- sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) = sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < )
13	1 2 3 4 5 6 7 8 11 12	ivthlem3	|- ( ph -> ( sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) e. ( A (,) B ) /\ ( F ` sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) ) = U ) )
14		fveqeq2	|- ( c = sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) -> ( ( F ` c ) = U <-> ( F ` sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) ) = U ) )
15	14	rspcev	|- ( ( sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) e. ( A (,) B ) /\ ( F ` sup ( { y e. ( A [,] B ) | ( F ` y ) <_ U } , RR , < ) ) = U ) -> E. c e. ( A (,) B ) ( F ` c ) = U )
16	13 15	syl	|- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )