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Mirrors > Home > MPE Home > Th. List > fviunfun | Structured version Visualization version GIF version |
Description: The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.) |
Ref | Expression |
---|---|
fviunfun.u | ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖) |
Ref | Expression |
---|---|
fviunfun | ⊢ ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6902 | . . . 4 ⊢ (𝑖 = 𝐽 → (𝐹‘𝑖) = (𝐹‘𝐽)) | |
2 | 1 | ssiun2s 5055 | . . 3 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)) |
3 | fviunfun.u | . . 3 ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖) | |
4 | 2, 3 | sseqtrrdi 4033 | . 2 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ 𝑈) |
5 | funssfv 6923 | . 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)) |
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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