Theorem fviunfun 7930

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 6885	. . . 4 ⊢ (𝑖 = 𝐽 → (𝐹‘𝑖) = (𝐹‘𝐽))
2	1	ssiun2s 5044	. . 3 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖))
3		fviunfun.u	. . 3 ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)
4	2, 3	sseqtrrdi 4028	. 2 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ 𝑈)
5		funssfv 6906	. 2 ⊢ ((Fun 𝑈 ∧ (𝐹‘𝐽) ⊆ 𝑈 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))
6	4, 5	syl3an2 1161	1 ⊢ ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3943  ∪ ciun 4990  dom cdm 5669  Fun wfun 6531  ‘cfv 6537

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 6489  df-fun 6539  df-fv 6545

This theorem is referenced by:  satefvfmla0  34937  satefvfmla1  34944