Theorem pjhval 29767

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 29766	. . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))))
2	1	fveq1d 6768	. 2 ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴))
3		eqeq1 2742	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ 𝑦)))
4	3	rexbidv 3224	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
5	4	riotabidv 7226	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))
6		eqid 2738	. . 3 ⊢ (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))
7		riotaex 7228	. . 3 ⊢ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ∈ V
8	5, 6, 7	fvmpt 6867	. 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 2798	1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ↦ cmpt 5156  ‘cfv 6426  ℩crio 7223  (class class class)co 7267   ℋchba 29289   +_ℎ cva 29290   C_ℋ cch 29299  ⊥cort 29300  proj_ℎcpjh 29307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-hilex 29369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-pjh 29765

This theorem is referenced by:  pjpreeq  29768