Theorem relfae 30310

Description: The 'almost everywhere' builder for functions produces relations. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Assertion

Ref	Expression
relfae	⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀))

Proof of Theorem relfae

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5247	. 2 ⊢ Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}
2		faeval 30309	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)})
3	2	releqd 5203	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (Rel (𝑅~ a.e.𝑀) ↔ Rel {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑𝑚 ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom 𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}))
4	1, 3	mpbiri 248	1 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → Rel (𝑅~ a.e.𝑀))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  {crab 2916  Vcvv 3200  ∪ cuni 4436   class class class wbr 4653  {copab 4712  dom cdm 5114  ran crn 5115  Rel wrel 5119  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  measurescmeas 30258  a.e.cae 30300  ~ a.e.cfae 30301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fae 30308

This theorem is referenced by: (None)