Theorem pjhfval 31991

Description: The value of the projection map. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
pjhfval	⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))))

Distinct variable group:   𝑥,𝑦,𝑧,𝐻

Proof of Theorem pjhfval

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
2		fveq2 6883	. . . . 5 ⊢ (ℎ = 𝐻 → (⊥‘ℎ) = (⊥‘𝐻))
3	2	rexeqdv 3321	. . . 4 ⊢ (ℎ = 𝐻 → (∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))
4	1, 3	riotaeqbidv 7378	. . 3 ⊢ (ℎ = 𝐻 → (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)) = (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))
5	4	mpteq2dv 5199	. 2 ⊢ (ℎ = 𝐻 → (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))))
6		df-pjh 31990	. 2 ⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
7		ax-hilex 31594	. . 3 ⊢ ℋ ∈ V
8	7	mptex 7227	. 2 ⊢ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))) ∈ V
9	5, 6, 8	fvmpt 6991	1 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ↦ cmpt 5186  ‘cfv 6537  ℩crio 7374  (class class class)co 7418   ℋchba 31514   +_ℎ cva 31515   C_ℋ cch 31524  ⊥cort 31525  proj_ℎcpjh 31532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-hilex 31594

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-pjh 31990

This theorem is used by:  pjhval  31992  pjfni  32296