Theorem pjhval 29180

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

Assertion

Ref	Expression
pjhval	⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐻   𝑥,𝐴,𝑦

Proof of Theorem pjhval

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjhfval 29179	. . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))))
2	1	fveq1d 6647	. 2 ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴))
3		eqeq1 2802	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ 𝑦)))
4	3	rexbidv 3256	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
5	4	riotabidv 7095	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
6		eqid 2798	. . 3 ⊢ (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))
7		riotaex 7097	. . 3 ⊢ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ∈ V
8	5, 6, 7	fvmpt 6745	. 2 ⊢ (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
9	2, 8	sylan9eq 2853	1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ↦ cmpt 5110  ‘cfv 6324  ℩crio 7092  (class class class)co 7135   ℋchba 28702   +_ℎ cva 28703   C_ℋ cch 28712  ⊥cort 28713  proj_ℎcpjh 28720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-pjh 29178

This theorem is referenced by:  pjpreeq  29181