Theorem hoeq 30122

Description: Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hoeq	⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈))

Distinct variable groups:   𝑥,𝑇   𝑥,𝑈

Proof of Theorem hoeq

Step	Hyp	Ref	Expression
1		ffn 6600	. 2 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
2		ffn 6600	. 2 ⊢ (𝑈: ℋ⟶ ℋ → 𝑈 Fn ℋ)
3		eqfnfv 6909	. . 3 ⊢ ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (𝑇 = 𝑈 ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥)))
4	3	bicomd 222	. 2 ⊢ ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈))
5	1, 2, 4	syl2an 596	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∀wral 3064   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433   ℋchba 29281

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-sep 5223  ax-nul 5230  ax-pr 5352

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-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  hoeqi  30123  homulid2  30162  homco1  30163  homulass  30164  hoadddi  30165  hoadddir  30166  homco2  30339