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Theorem bj-charfundcALT 13371
Description: Alternate proof of bj-charfundc 13370. It was expected to be much shorter since it uses bj-charfun 13369 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
bj-charfundc.1 (𝜑𝐹 = (𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)))
bj-charfundc.dc (𝜑 → ∀𝑥𝑋 DECID 𝑥𝐴)
Assertion
Ref Expression
bj-charfundcALT (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = ∅)))
Distinct variable groups:   𝜑,𝑥   𝑥,𝑋   𝑥,𝐴   𝑥,𝐹

Proof of Theorem bj-charfundcALT
StepHypRef Expression
1 bj-charfundc.1 . . 3 (𝜑𝐹 = (𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)))
21bj-charfun 13369 . 2 (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))):((𝑋𝐴) ∪ (𝑋𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = ∅)))
3 difin 3344 . . . . . . . . . . . 12 (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐴)
43eqcomi 2161 . . . . . . . . . . 11 (𝑋𝐴) = (𝑋 ∖ (𝑋𝐴))
54a1i 9 . . . . . . . . . 10 (𝜑 → (𝑋𝐴) = (𝑋 ∖ (𝑋𝐴)))
65uneq2d 3261 . . . . . . . . 9 (𝜑 → ((𝑋𝐴) ∪ (𝑋𝐴)) = ((𝑋𝐴) ∪ (𝑋 ∖ (𝑋𝐴))))
7 inss1 3327 . . . . . . . . . . 11 (𝑋𝐴) ⊆ 𝑋
87a1i 9 . . . . . . . . . 10 (𝜑 → (𝑋𝐴) ⊆ 𝑋)
9 bj-charfundc.dc . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋 DECID 𝑥𝐴)
10 elin 3290 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝑋𝐴) ↔ (𝑥𝑋𝑥𝐴))
1110baibr 906 . . . . . . . . . . . . 13 (𝑥𝑋 → (𝑥𝐴𝑥 ∈ (𝑋𝐴)))
1211dcbid 824 . . . . . . . . . . . 12 (𝑥𝑋 → (DECID 𝑥𝐴DECID 𝑥 ∈ (𝑋𝐴)))
1312ralbiia 2471 . . . . . . . . . . 11 (∀𝑥𝑋 DECID 𝑥𝐴 ↔ ∀𝑥𝑋 DECID 𝑥 ∈ (𝑋𝐴))
149, 13sylib 121 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋 DECID 𝑥 ∈ (𝑋𝐴))
15 undifdcss 6864 . . . . . . . . . 10 (𝑋 = ((𝑋𝐴) ∪ (𝑋 ∖ (𝑋𝐴))) ↔ ((𝑋𝐴) ⊆ 𝑋 ∧ ∀𝑥𝑋 DECID 𝑥 ∈ (𝑋𝐴)))
168, 14, 15sylanbrc 414 . . . . . . . . 9 (𝜑𝑋 = ((𝑋𝐴) ∪ (𝑋 ∖ (𝑋𝐴))))
176, 16eqtr4d 2193 . . . . . . . 8 (𝜑 → ((𝑋𝐴) ∪ (𝑋𝐴)) = 𝑋)
1817reseq2d 4865 . . . . . . 7 (𝜑 → (𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))) = (𝐹𝑋))
19 ssidd 3149 . . . . . . . . 9 (𝜑𝑋𝑋)
2019resmptd 4916 . . . . . . . 8 (𝜑 → ((𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)) ↾ 𝑋) = (𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)))
211reseq1d 4864 . . . . . . . 8 (𝜑 → (𝐹𝑋) = ((𝑥𝑋 ↦ if(𝑥𝐴, 1o, ∅)) ↾ 𝑋))
2220, 21, 13eqtr4d 2200 . . . . . . 7 (𝜑 → (𝐹𝑋) = 𝐹)
2318, 22eqtrd 2190 . . . . . 6 (𝜑 → (𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))) = 𝐹)
2423, 17feq12d 5308 . . . . 5 (𝜑 → ((𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))):((𝑋𝐴) ∪ (𝑋𝐴))⟶2o𝐹:𝑋⟶2o))
2524biimpd 143 . . . 4 (𝜑 → ((𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))):((𝑋𝐴) ∪ (𝑋𝐴))⟶2o𝐹:𝑋⟶2o))
2625adantld 276 . . 3 (𝜑 → ((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))):((𝑋𝐴) ∪ (𝑋𝐴))⟶2o) → 𝐹:𝑋⟶2o))
2726anim1d 334 . 2 (𝜑 → (((𝐹:𝑋⟶𝒫 1o ∧ (𝐹 ↾ ((𝑋𝐴) ∪ (𝑋𝐴))):((𝑋𝐴) ∪ (𝑋𝐴))⟶2o) ∧ (∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = ∅)) → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = ∅))))
282, 27mpd 13 1 (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋𝐴)(𝐹𝑥) = ∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 820   = wceq 1335  wcel 2128  wral 2435  cdif 3099  cun 3100  cin 3101  wss 3102  c0 3394  ifcif 3505  𝒫 cpw 3543  cmpt 4025  cres 4587  wf 5165  cfv 5169  1oc1o 6353  2oc2o 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-1o 6360  df-2o 6361
This theorem is referenced by: (None)
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