Theorem pjhfval 31226

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 22	. . . 4 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
2		fveq2 6902	. . . . 5 ⊢ (ℎ = 𝐻 → (⊥‘ℎ) = (⊥‘𝐻))
3	2	rexeqdv 3324	. . . 4 ⊢ (ℎ = 𝐻 → (∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))
4	1, 3	riotaeqbidv 7385	. . 3 ⊢ (ℎ = 𝐻 → (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)) = (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦)))
5	4	mpteq2dv 5254	. 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 31225	. 2 ⊢ projℎ = (ℎ ∈ Cℋ ↦ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦))))
7		ax-hilex 30829	. . 3 ⊢ ℋ ∈ V
8	7	mptex 7241	. 2 ⊢ (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))) ∈ V
9	5, 6, 8	fvmpt 7010	1 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑧 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑥 = (𝑧 +ℎ 𝑦))))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∃wrex 3067   ↦ cmpt 5235  ‘cfv 6553  ℩crio 7381  (class class class)co 7426   ℋchba 30749   +_ℎ cva 30750   C_ℋ cch 30759  ⊥cort 30760  proj_ℎcpjh 30767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-hilex 30829

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-pjh 31225

This theorem is referenced by:  pjhval  31227  pjfni  31531