MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fviunfun Structured version   Visualization version   GIF version

Theorem fviunfun 7927
Description: The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.)
Hypothesis
Ref Expression
fviunfun.u 𝑈 = 𝑖𝐼 (𝐹𝑖)
Assertion
Ref Expression
fviunfun ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
Distinct variable groups:   𝑖,𝐹   𝑖,𝐼   𝑖,𝐽
Allowed substitution hints:   𝑈(𝑖)   𝑋(𝑖)

Proof of Theorem fviunfun
StepHypRef Expression
1 fveq2 6884 . . . 4 (𝑖 = 𝐽 → (𝐹𝑖) = (𝐹𝐽))
21ssiun2s 5044 . . 3 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑖𝐼 (𝐹𝑖))
3 fviunfun.u . . 3 𝑈 = 𝑖𝐼 (𝐹𝑖)
42, 3sseqtrrdi 4028 . 2 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑈)
5 funssfv 6905 . 2 ((Fun 𝑈 ∧ (𝐹𝐽) ⊆ 𝑈𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
64, 5syl3an2 1161 1 ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wss 3943   ciun 4990  dom cdm 5669  Fun wfun 6530  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544
This theorem is referenced by:  satefvfmla0  34936  satefvfmla1  34943
  Copyright terms: Public domain W3C validator