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Theorem fviunfun 7968
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 6907 . . . 4 (𝑖 = 𝐽 → (𝐹𝑖) = (𝐹𝐽))
21ssiun2s 5053 . . 3 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑖𝐼 (𝐹𝑖))
3 fviunfun.u . . 3 𝑈 = 𝑖𝐼 (𝐹𝑖)
42, 3sseqtrrdi 4047 . 2 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑈)
5 funssfv 6928 . 2 ((Fun 𝑈 ∧ (𝐹𝐽) ⊆ 𝑈𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
64, 5syl3an2 1163 1 ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wss 3963   ciun 4996  dom cdm 5689  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  satefvfmla0  35403  satefvfmla1  35410
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