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Mirrors > Home > MPE Home > Th. List > fvun1 | Structured version Visualization version GIF version |
Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
fvun1 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 5902 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1075 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐹) |
3 | fnfun 5902 | . . . 4 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺) | |
4 | 3 | 3ad2ant2 1076 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → Fun 𝐺) |
5 | fndm 5904 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
6 | fndm 5904 | . . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
7 | 5, 6 | ineqan12d 3778 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
8 | 7 | eqeq1d 2612 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) |
9 | 8 | biimprd 237 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (dom 𝐹 ∩ dom 𝐺) = ∅)) |
10 | 9 | adantrd 483 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → (dom 𝐹 ∩ dom 𝐺) = ∅)) |
11 | 10 | 3impia 1253 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (dom 𝐹 ∩ dom 𝐺) = ∅) |
12 | fvun 6178 | . . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋))) | |
13 | 2, 4, 11, 12 | syl21anc 1317 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = ((𝐹‘𝑋) ∪ (𝐺‘𝑋))) |
14 | disjel 3975 | . . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ 𝐵) | |
15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ 𝐵) |
16 | 6 | eleq2d 2673 | . . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐵)) |
17 | 16 | adantr 480 | . . . . . . 7 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐵)) |
18 | 15, 17 | mtbird 314 | . . . . . 6 ⊢ ((𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ dom 𝐺) |
19 | 18 | 3adant1 1072 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ¬ 𝑋 ∈ dom 𝐺) |
20 | ndmfv 6128 | . . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐺 → (𝐺‘𝑋) = ∅) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) = ∅) |
22 | 21 | uneq2d 3729 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = ((𝐹‘𝑋) ∪ ∅)) |
23 | un0 3919 | . . 3 ⊢ ((𝐹‘𝑋) ∪ ∅) = (𝐹‘𝑋) | |
24 | 22, 23 | syl6eq 2660 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑋) ∪ (𝐺‘𝑋)) = (𝐹‘𝑋)) |
25 | 13, 24 | eqtrd 2644 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: fvun2 6180 enfixsn 7954 hashf1lem1 13096 xpsc0 16043 ptunhmeo 21421 axlowdimlem6 25627 axlowdimlem8 25629 axlowdimlem11 25632 constr3lem4 26175 vdgrun 26428 vdgrfiun 26429 isoun 28862 sseqfv1 29778 cvmliftlem5 30525 fullfunfv 31224 finixpnum 32564 poimirlem1 32580 poimirlem2 32581 poimirlem3 32582 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem11 32590 poimirlem12 32591 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem22 32601 poimirlem23 32602 poimirlem28 32607 vtxdun 40696 aacllem 42356 |
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