Theorem pjval 20827

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 6756	. . 3 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 7273	. 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 20826	. 2 ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
8		ovex 7288	. 2 ⊢ (𝑇𝑃( ⊥ ‘𝑇)) ∈ V
9	3, 7, 8	fvmpt 6857	1 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  proj₁cpj1 19155  ocvcocv 20777  projcpj 20817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-pj 20820

This theorem is referenced by:  pjf  20830  pjf2  20831  pjfo  20832