Theorem fviunfun 7899

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 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)

Assertion

Ref	Expression
fviunfun	⊢ ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))

Distinct variable groups:   𝑖,𝐹   𝑖,𝐼   𝑖,𝐽

Allowed substitution hints:   𝑈(𝑖)   𝑋(𝑖)

Proof of Theorem fviunfun

Step	Hyp	Ref	Expression
1		fveq2 6842	. . . 4 ⊢ (𝑖 = 𝐽 → (𝐹‘𝑖) = (𝐹‘𝐽))
2	1	ssiun2s 5006	. . 3 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖))
3		fviunfun.u	. . 3 ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)
4	2, 3	sseqtrrdi 3977	. 2 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ 𝑈)
5		funssfv 6863	. 2 ⊢ ((Fun 𝑈 ∧ (𝐹‘𝐽) ⊆ 𝑈 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))
6	4, 5	syl3an2 1165	1 ⊢ ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∪ ciun 4948  dom cdm 5632  Fun wfun 6494  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508

This theorem is referenced by:  satefvfmla0  35634  satefvfmla1  35641