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Theorem fviunfun 7878
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 6843 . . . 4 (𝑖 = 𝐽 → (𝐹𝑖) = (𝐹𝐽))
21ssiun2s 5009 . . 3 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑖𝐼 (𝐹𝑖))
3 fviunfun.u . . 3 𝑈 = 𝑖𝐼 (𝐹𝑖)
42, 3sseqtrrdi 3996 . 2 (𝐽𝐼 → (𝐹𝐽) ⊆ 𝑈)
5 funssfv 6864 . 2 ((Fun 𝑈 ∧ (𝐹𝐽) ⊆ 𝑈𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
64, 5syl3an2 1165 1 ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wss 3911   ciun 4955  dom cdm 5634  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505
This theorem is referenced by:  satefvfmla0  34069  satefvfmla1  34076
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