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Theorem fsuppssindlem2 42602
Description: Lemma for fsuppssind 42603. 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 6905 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
21ifeq1d 4545 . . . . 5 (𝑓 = 𝐹 → if(𝑥𝑆, (𝑓𝑥), 0 ) = if(𝑥𝑆, (𝐹𝑥), 0 ))
32mpteq2dv 5244 . . . 4 (𝑓 = 𝐹 → (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) = (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )))
43eleq1d 2826 . . 3 (𝑓 = 𝐹 → ((𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻))
54elrab 3692 . 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 5324 . . . . 5 (𝜑𝑆 ∈ V)
106, 9elmapd 8880 . . . 4 (𝜑 → (𝐹 ∈ (𝐵m 𝑆) ↔ 𝐹:𝑆𝐵))
1110anbi1d 631 . . 3 (𝜑 → ((𝐹 ∈ (𝐵m 𝑆) ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻)))
12 partfun 6715 . . . . . 6 (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) = ((𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) ∪ (𝑥 ∈ (𝐼𝑆) ↦ 0 ))
13 sseqin2 4223 . . . . . . . . . . 11 (𝑆𝐼 ↔ (𝐼𝑆) = 𝑆)
148, 13sylib 218 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) = 𝑆)
1514mpteq1d 5237 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = (𝑥𝑆 ↦ (𝐹𝑥)))
1615adantr 480 . . . . . . . 8 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = (𝑥𝑆 ↦ (𝐹𝑥)))
17 simpr 484 . . . . . . . . 9 ((𝜑𝐹:𝑆𝐵) → 𝐹:𝑆𝐵)
1817feqmptd 6977 . . . . . . . 8 ((𝜑𝐹:𝑆𝐵) → 𝐹 = (𝑥𝑆 ↦ (𝐹𝑥)))
1916, 18eqtr4d 2780 . . . . . . 7 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) = 𝐹)
20 fconstmpt 5747 . . . . . . . . 9 ((𝐼𝑆) × { 0 }) = (𝑥 ∈ (𝐼𝑆) ↦ 0 )
2120eqcomi 2746 . . . . . . . 8 (𝑥 ∈ (𝐼𝑆) ↦ 0 ) = ((𝐼𝑆) × { 0 })
2221a1i 11 . . . . . . 7 ((𝜑𝐹:𝑆𝐵) → (𝑥 ∈ (𝐼𝑆) ↦ 0 ) = ((𝐼𝑆) × { 0 }))
2319, 22uneq12d 4169 . . . . . 6 ((𝜑𝐹:𝑆𝐵) → ((𝑥 ∈ (𝐼𝑆) ↦ (𝐹𝑥)) ∪ (𝑥 ∈ (𝐼𝑆) ↦ 0 )) = (𝐹 ∪ ((𝐼𝑆) × { 0 })))
2412, 23eqtrid 2789 . . . . 5 ((𝜑𝐹:𝑆𝐵) → (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) = (𝐹 ∪ ((𝐼𝑆) × { 0 })))
2524eleq1d 2826 . . . 4 ((𝜑𝐹:𝑆𝐵) → ((𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻 ↔ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))
2625pm5.32da 579 . . 3 (𝜑 → ((𝐹:𝑆𝐵 ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
2711, 26bitrd 279 . 2 (𝜑 → ((𝐹 ∈ (𝐵m 𝑆) ∧ (𝑥𝐼 ↦ if(𝑥𝑆, (𝐹𝑥), 0 )) ∈ 𝐻) ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
285, 27bitrid 283 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 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  ifcif 4525  {csn 4626  cmpt 5225   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868
This theorem is referenced by:  fsuppssind  42603
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