Theorem ppival 24853

Description: Value of the prime-counting function pi. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
ppival	⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))

Proof of Theorem ppival

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 ⊢ (𝑥 = 𝐴 → (0[,]𝑥) = (0[,]𝐴))
2	1	ineq1d 3813	. . 3 ⊢ (𝑥 = 𝐴 → ((0[,]𝑥) ∩ ℙ) = ((0[,]𝐴) ∩ ℙ))
3	2	fveq2d 6195	. 2 ⊢ (𝑥 = 𝐴 → (#‘((0[,]𝑥) ∩ ℙ)) = (#‘((0[,]𝐴) ∩ ℙ)))
4		df-ppi 24826	. 2 ⊢ π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩ ℙ)))
5		fvex 6201	. 2 ⊢ (#‘((0[,]𝐴) ∩ ℙ)) ∈ V
6	3, 4, 5	fvmpt 6282	1 ⊢ (𝐴 ∈ ℝ → (π‘𝐴) = (#‘((0[,]𝐴) ∩ ℙ)))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ∩ cin 3573  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  [,]cicc 12178  #chash 13117  ℙcprime 15385  πcppi 24820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-ppi 24826

This theorem is referenced by:  ppival2  24854  ppival2g  24855  ppifl  24886  ppiwordi  24888  chtleppi  24935