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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 6663 | . . . 4 ⊢ (𝑖 = 𝐽 → (𝐹‘𝑖) = (𝐹‘𝐽)) | |
2 | 1 | ssiun2s 4965 | . . 3 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)) |
3 | fviunfun.u | . . 3 ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖) | |
4 | 2, 3 | sseqtrrdi 4011 | . 2 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ 𝑈) |
5 | funssfv 6684 | . 2 ⊢ ((Fun 𝑈 ∧ (𝐹‘𝐽) ⊆ 𝑈 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋)) | |
6 | 4, 5 | syl3an2 1159 | 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 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 ∪ ciun 4912 dom cdm 5548 Fun wfun 6342 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: satefvfmla0 32684 satefvfmla1 32691 |
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