Theorem pjval 20054

Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypotheses

Ref	Expression
pjfval2.o	⊢ ⊥ = (ocv‘𝑊)
pjfval2.p	⊢ 𝑃 = (proj1‘𝑊)
pjfval2.k	⊢ 𝐾 = (proj‘𝑊)

Assertion

Ref	Expression
pjval	⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇)))

Proof of Theorem pjval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑥 = 𝑇 → 𝑥 = 𝑇)
2		fveq2 6191	. . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 6668	. 2 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4		pjfval2.o	. . 3 ⊢ ⊥ = (ocv‘𝑊)
5		pjfval2.p	. . 3 ⊢ 𝑃 = (proj1‘𝑊)
6		pjfval2.k	. . 3 ⊢ 𝐾 = (proj‘𝑊)
7	4, 5, 6	pjfval2 20053	. 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
8		ovex 6678	. 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V
9	3, 7, 8	fvmpt 6282	1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇)))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  proj₁cpj1 18050  ocvcocv 20004  projcpj 20044

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-pw 4160  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-pj 20047

This theorem is referenced by:  pjf  20057  pjf2  20058  pjfo  20059