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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 6843 | . . . 4 ⊢ (𝑖 = 𝐽 → (𝐹‘𝑖) = (𝐹‘𝐽)) | |
2 | 1 | ssiun2s 5009 | . . 3 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)) |
3 | fviunfun.u | . . 3 ⊢ 𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖) | |
4 | 2, 3 | sseqtrrdi 3996 | . 2 ⊢ (𝐽 ∈ 𝐼 → (𝐹‘𝐽) ⊆ 𝑈) |
5 | funssfv 6864 | . 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)) |
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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