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Theorem bj-charfundcALT 14531
Description: Alternate proof of bj-charfundc 14530. It was expected to be much shorter since it uses bj-charfun 14529 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 14529 . 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 3372 . . . . . . . . . . . 12 (𝑋 βˆ– (𝑋 ∩ 𝐴)) = (𝑋 βˆ– 𝐴)
43eqcomi 2181 . . . . . . . . . . 11 (𝑋 βˆ– 𝐴) = (𝑋 βˆ– (𝑋 ∩ 𝐴))
54a1i 9 . . . . . . . . . 10 (πœ‘ β†’ (𝑋 βˆ– 𝐴) = (𝑋 βˆ– (𝑋 ∩ 𝐴)))
65uneq2d 3289 . . . . . . . . 9 (πœ‘ β†’ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴)) = ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– (𝑋 ∩ 𝐴))))
7 inss1 3355 . . . . . . . . . . 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 3318 . . . . . . . . . . . . . 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 2491 . . . . . . . . . . 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 6921 . . . . . . . . . 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 2213 . . . . . . . 8 (πœ‘ β†’ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴)) = 𝑋)
1817reseq2d 4907 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))) = (𝐹 β†Ύ 𝑋))
19 ssidd 3176 . . . . . . . . 9 (πœ‘ β†’ 𝑋 βŠ† 𝑋)
2019resmptd 4958 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)) β†Ύ 𝑋) = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)))
211reseq1d 4906 . . . . . . . 8 (πœ‘ β†’ (𝐹 β†Ύ 𝑋) = ((π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ 𝐴, 1o, βˆ…)) β†Ύ 𝑋))
2220, 21, 13eqtr4d 2220 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ 𝑋) = 𝐹)
2318, 22eqtrd 2210 . . . . . 6 (πœ‘ β†’ (𝐹 β†Ύ ((𝑋 ∩ 𝐴) βˆͺ (𝑋 βˆ– 𝐴))) = 𝐹)
2423, 17feq12d 5355 . . . . 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 2148  βˆ€wral 2455   βˆ– cdif 3126   βˆͺ cun 3127   ∩ cin 3128   βŠ† wss 3129  βˆ…c0 3422  ifcif 3534  π’« cpw 3575   ↦ cmpt 4064   β†Ύ cres 4628  βŸΆwf 5212  β€˜cfv 5216  1oc1o 6409  2oc2o 6410
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-1o 6416  df-2o 6417
This theorem is referenced by: (None)
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