Theorem hoeq 31831

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 6668	. 2 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
2		ffn 6668	. 2 ⊢ (𝑈: ℋ⟶ ℋ → 𝑈 Fn ℋ)
3		eqfnfv 6983	. . 3 ⊢ ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (𝑇 = 𝑈 ↔ ∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥)))
4	3	bicomd 223	. 2 ⊢ ((𝑇 Fn ℋ ∧ 𝑈 Fn ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈))
5	1, 2, 4	syl2an 597	1 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑇‘𝑥) = (𝑈‘𝑥) ↔ 𝑇 = 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∀wral 3051   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498   ℋchba 30990

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  hoeqi  31832  homullid  31871  homco1  31872  homulass  31873  hoadddi  31874  hoadddir  31875  homco2  32048