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Theorem fviunfun 7954
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 6902 . . . 4 (𝑖 = 𝐽 → (𝐹𝑖) = (𝐹𝐽))
21ssiun2s 5055 . . 3 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑖𝐼 (𝐹𝑖))
3 fviunfun.u . . 3 𝑈 = 𝑖𝐼 (𝐹𝑖)
42, 3sseqtrrdi 4033 . 2 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑈)
5 funssfv 6923 . 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 3949   ciun 5000  dom cdm 5682  Fun wfun 6547  cfv 6553
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561
This theorem is referenced by:  satefvfmla0  35061  satefvfmla1  35068
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