Mathbox for Steven Nguyen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsuppssindlem2 Structured version   Visualization version   GIF version

Theorem fsuppssindlem2 39542
 Description: Lemma for fsuppssind 39543. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
Hypotheses
Ref Expression
fsuppssindlem2.b (𝜑𝐵𝑊)
fsuppssindlem2.v (𝜑𝐼𝑉)
fsuppssindlem2.s (𝜑𝑆𝐼)
Assertion
Ref Expression
fsuppssindlem2 (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
Distinct variable groups:   𝑓,𝐼,𝑥   𝑆,𝑓,𝑥   𝑓,𝐹,𝑥   0 ,𝑓,𝑥   𝑓,𝐻   𝐵,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥)   𝐻(𝑥)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)

Proof of Theorem fsuppssindlem2
StepHypRef Expression
1 fveq1 6651 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
21ifeq1d 4445 . . . . 5 (𝑓 = 𝐹 → if(𝑥𝑆, (𝑓𝑥), 0 ) = if(𝑥𝑆, (𝐹𝑥), 0 ))
32mpteq2dv 5129 . . . 4 (𝑓 = 𝐹 → (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) = (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )))
43eleq1d 2874 . . 3 (𝑓 = 𝐹 → ((𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻))
54elrab 3629 . 2 (𝐹 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻} ↔ (𝐹 ∈ (𝐵m 𝑆) ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻))
6 fsuppssindlem2.b . . . . 5 (𝜑𝐵𝑊)
7 fsuppssindlem2.v . . . . . 6 (𝜑𝐼𝑉)
8 fsuppssindlem2.s . . . . . 6 (𝜑𝑆𝐼)
97, 8ssexd 5195 . . . . 5 (𝜑𝑆 ∈ V)
106, 9elmapd 8418 . . . 4 (𝜑 → (𝐹 ∈ (𝐵m 𝑆) ↔ 𝐹:𝑆𝐵))
1110anbi1d 632 . . 3 (𝜑 → ((𝐹 ∈ (𝐵m 𝑆) ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻)))
12 partfun 6472 . . . . . 6 (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) = ((𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) ∪ (𝑥 ∈ (𝐼𝑆) ↦ 0 ))
13 sseqin2 4144 . . . . . . . . . . 11 (𝑆𝐼 ↔ (𝐼𝑆) = 𝑆)
148, 13sylib 221 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) = 𝑆)
1514mpteq1d 5122 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = (𝑥𝑆 ↦ (𝐹𝑥)))
1615adantr 484 . . . . . . . 8 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = (𝑥𝑆 ↦ (𝐹𝑥)))
17 simpr 488 . . . . . . . . 9 ((𝜑𝐹:𝑆𝐵) → 𝐹:𝑆𝐵)
1817feqmptd 6715 . . . . . . . 8 ((𝜑𝐹:𝑆𝐵) → 𝐹 = (𝑥𝑆 ↦ (𝐹𝑥)))
1916, 18eqtr4d 2836 . . . . . . 7 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = 𝐹)
20 fconstmpt 5581 . . . . . . . . 9 ((𝐼𝑆) × { 0 }) = (𝑥 ∈ (𝐼𝑆) ↦ 0 )
2120eqcomi 2807 . . . . . . . 8 (𝑥 ∈ (𝐼𝑆) ↦ 0 ) = ((𝐼𝑆) × { 0 })
2221a1i 11 . . . . . . 7 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ 0 ) = ((𝐼𝑆) × { 0 }))
2319, 22uneq12d 4093 . . . . . 6 ((𝜑𝐹:𝑆𝐵) → ((𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) ∪ (𝑥 ∈ (𝐼𝑆) ↦ 0 )) = (𝐹 ∪ ((𝐼𝑆) × { 0 })))
2412, 23syl5eq 2845 . . . . 5 ((𝜑𝐹:𝑆𝐵) → (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) = (𝐹 ∪ ((𝐼𝑆) × { 0 })))
2524eleq1d 2874 . . . 4 ((𝜑𝐹:𝑆𝐵) → ((𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻 ↔ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))
2625pm5.32da 582 . . 3 (𝜑 → ((𝐹:𝑆𝐵 ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
2711, 26bitrd 282 . 2 (𝜑 → ((𝐹 ∈ (𝐵m 𝑆) ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
285, 27syl5bb 286 1 (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3879   ∪ cun 3880   ∩ cin 3881   ⊆ wss 3882  ifcif 4427  {csn 4527   ↦ cmpt 5113   × cxp 5520  ⟶wf 6325  ‘cfv 6329  (class class class)co 7142   ↑m cmap 8404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-map 8406 This theorem is referenced by:  fsuppssind  39543
 Copyright terms: Public domain W3C validator