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Theorem bj-charfundcALT 14121
Description: Alternate proof of bj-charfundc 14120. It was expected to be much shorter since it uses bj-charfun 14119 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 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))
bj-charfundc.dc (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)
Assertion
Ref Expression
bj-charfundcALT (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝑋   π‘₯,𝐴   π‘₯,𝐹

Proof of Theorem bj-charfundcALT
StepHypRef Expression
1 bj-charfundc.1 . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))
21bj-charfun 14119 . 2 (πœ‘ β†’ ((𝐹:π‘‹βŸΆπ’« 1o ∧ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o) ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
3 difin 3370 . . . . . . . . . . . 12 (𝑋 βˆ– (𝑋 ∩ 𝐴)) = (𝑋 βˆ– 𝐴)
43eqcomi 2179 . . . . . . . . . . 11 (𝑋 βˆ– 𝐴) = (𝑋 βˆ– (𝑋 ∩ 𝐴))
54a1i 9 . . . . . . . . . 10 (πœ‘ β†’ (𝑋 βˆ– 𝐴) = (𝑋 βˆ– (𝑋 ∩ 𝐴)))
65uneq2d 3287 . . . . . . . . 9 (πœ‘ β†’ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴)) = ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– (𝑋 ∩ 𝐴))))
7 inss1 3353 . . . . . . . . . . 11 (𝑋 ∩ 𝐴) βŠ† 𝑋
87a1i 9 . . . . . . . . . 10 (πœ‘ β†’ (𝑋 ∩ 𝐴) βŠ† 𝑋)
9 bj-charfundc.dc . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴)
10 elin 3316 . . . . . . . . . . . . . 14 (π‘₯ ∈ (𝑋 ∩ 𝐴) ↔ (π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ 𝐴))
1110baibr 920 . . . . . . . . . . . . 13 (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ (𝑋 ∩ 𝐴)))
1211dcbid 838 . . . . . . . . . . . 12 (π‘₯ ∈ 𝑋 β†’ (DECID π‘₯ ∈ 𝐴 ↔ DECID π‘₯ ∈ (𝑋 ∩ 𝐴)))
1312ralbiia 2489 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ 𝐴 ↔ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ (𝑋 ∩ 𝐴))
149, 13sylib 122 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ (𝑋 ∩ 𝐴))
15 undifdcss 6912 . . . . . . . . . 10 (𝑋 = ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– (𝑋 ∩ 𝐴))) ↔ ((𝑋 ∩ 𝐴) βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 DECID π‘₯ ∈ (𝑋 ∩ 𝐴)))
168, 14, 15sylanbrc 417 . . . . . . . . 9 (πœ‘ β†’ 𝑋 = ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– (𝑋 ∩ 𝐴))))
176, 16eqtr4d 2211 . . . . . . . 8 (πœ‘ β†’ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴)) = 𝑋)
1817reseq2d 4900 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))) = (𝐹 β†Ύ 𝑋))
19 ssidd 3174 . . . . . . . . 9 (πœ‘ β†’ 𝑋 βŠ† 𝑋)
2019resmptd 4951 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)) β†Ύ 𝑋) = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))
211reseq1d 4899 . . . . . . . 8 (πœ‘ β†’ (𝐹 β†Ύ 𝑋) = ((π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)) β†Ύ 𝑋))
2220, 21, 13eqtr4d 2218 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ 𝑋) = 𝐹)
2318, 22eqtrd 2208 . . . . . 6 (πœ‘ β†’ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))) = 𝐹)
2423, 17feq12d 5347 . . . . 5 (πœ‘ β†’ ((𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o ↔ 𝐹:π‘‹βŸΆ2o))
2524biimpd 144 . . . 4 (πœ‘ β†’ ((𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o β†’ 𝐹:π‘‹βŸΆ2o))
2625adantld 278 . . 3 (πœ‘ β†’ ((𝐹:π‘‹βŸΆπ’« 1o ∧ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o) β†’ 𝐹:π‘‹βŸΆ2o))
2726anim1d 336 . 2 (πœ‘ β†’ (((𝐹:π‘‹βŸΆπ’« 1o ∧ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))):((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))⟢2o) ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)) β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…))))
282, 27mpd 13 1 (πœ‘ β†’ (𝐹:π‘‹βŸΆ2o ∧ (βˆ€π‘₯ ∈ (𝑋 ∩ 𝐴)(πΉβ€˜π‘₯) = 1o ∧ βˆ€π‘₯ ∈ (𝑋 βˆ– 𝐴)(πΉβ€˜π‘₯) = βˆ…)))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104  DECID wdc 834   = wceq 1353   ∈ wcel 2146  βˆ€wral 2453   βˆ– cdif 3124   βˆͺ cun 3125   ∩ cin 3126   βŠ† wss 3127  βˆ…c0 3420  ifcif 3532  π’« cpw 3572   ↦ cmpt 4059   β†Ύ cres 4622  βŸΆwf 5204  β€˜cfv 5208  1oc1o 6400  2oc2o 6401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-1o 6407  df-2o 6408
This theorem is referenced by: (None)
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