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Theorem dif1enOLD 9187
Description: Obsolete version of dif1en 9185 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dif1enOLD ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1enOLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 bren 8974 . . 3 (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀)
2 19.41v 1946 . . . . 5 (∃𝑓(𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ ω)) ↔ (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ ω)))
3 3anass 1093 . . . . . 6 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) ↔ (𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ ω)))
43exbii 1843 . . . . 5 (∃𝑓(𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) ↔ ∃𝑓(𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ ω)))
5 3anass 1093 . . . . 5 ((∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) ↔ (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀 ∧ (𝑋𝐴𝑀 ∈ ω)))
62, 4, 53bitr4i 303 . . . 4 (∃𝑓(𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) ↔ (∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω))
7 sucidg 6450 . . . . . . . . 9 (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
8 f1ocnvdm 7294 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
983adant2 1129 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (𝑓𝑀) ∈ 𝐴)
10 f1ofvswap 7315 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴 ∧ (𝑓𝑀) ∈ 𝐴) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
119, 10syld3an3 1407 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
12 f1ocnvfv2 7286 . . . . . . . . . . . . . . . . 17 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (𝑓‘(𝑓𝑀)) = 𝑀)
1312opeq2d 4881 . . . . . . . . . . . . . . . 16 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ⟨𝑋, (𝑓‘(𝑓𝑀))⟩ = ⟨𝑋, 𝑀⟩)
1413preq1d 4744 . . . . . . . . . . . . . . 15 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} = {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
1514uneq2d 4162 . . . . . . . . . . . . . 14 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) = ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
1615f1oeq1d 6834 . . . . . . . . . . . . 13 ((𝑓:𝐴1-1-onto→suc 𝑀𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
17163adant2 1129 . . . . . . . . . . . 12 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, (𝑓‘(𝑓𝑀))⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 ↔ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀))
1811, 17mpbid 231 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
19 f1ofun 6841 . . . . . . . . . . 11 (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀 → Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
20 opex 5466 . . . . . . . . . . . . . 14 𝑋, 𝑀⟩ ∈ V
2120prid1 4767 . . . . . . . . . . . . 13 𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}
22 elun2 4177 . . . . . . . . . . . . 13 (⟨𝑋, 𝑀⟩ ∈ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} → ⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}))
2321, 22ax-mp 5 . . . . . . . . . . . 12 𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})
24 funopfv 6949 . . . . . . . . . . . 12 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (⟨𝑋, 𝑀⟩ ∈ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀))
2523, 24mpi 20 . . . . . . . . . . 11 (Fun ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
2618, 19, 253syl 18 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀)
27 simp2 1135 . . . . . . . . . . 11 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → 𝑋𝐴)
28 f1ocnvfv 7287 . . . . . . . . . . 11 ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀𝑋𝐴) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
2918, 27, 28syl2anc 583 . . . . . . . . . 10 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑋) = 𝑀 → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋))
3026, 29mpd 15 . . . . . . . . 9 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ suc 𝑀) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋)
317, 30syl3an3 1163 . . . . . . . 8 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀) = 𝑋)
3231sneqd 4641 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)} = {𝑋})
3332difeq2d 4120 . . . . . 6 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) = (𝐴 ∖ {𝑋}))
34 vex 3475 . . . . . . . . 9 𝑓 ∈ V
3534resex 6033 . . . . . . . 8 (𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∈ V
36 prex 5434 . . . . . . . 8 {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩} ∈ V
3735, 36unex 7748 . . . . . . 7 ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V
38 simp3 1136 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → 𝑀 ∈ ω)
397, 18syl3an3 1163 . . . . . . 7 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀)
40 dif1enlemOLD 9182 . . . . . . 7 ((((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}) ∈ V ∧ 𝑀 ∈ ω ∧ ((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩}):𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
4137, 38, 39, 40mp3an2i 1463 . . . . . 6 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {(((𝑓 ↾ (𝐴 ∖ {𝑋, (𝑓𝑀)})) ∪ {⟨𝑋, 𝑀⟩, ⟨(𝑓𝑀), (𝑓𝑋)⟩})‘𝑀)}) ≈ 𝑀)
4233, 41eqbrtrrd 5172 . . . . 5 ((𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
4342exlimiv 1926 . . . 4 (∃𝑓(𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
446, 43sylbir 234 . . 3 ((∃𝑓 𝑓:𝐴1-1-onto→suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
451, 44syl3an1b 1401 . 2 ((𝐴 ≈ suc 𝑀𝑋𝐴𝑀 ∈ ω) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
46453comr 1123 1 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  Vcvv 3471  cdif 3944  cun 3945  {csn 4629  {cpr 4631  cop 4635   class class class wbr 5148  ccnv 5677  cres 5680  suc csuc 6371  Fun wfun 6542  1-1-ontowf1o 6547  cfv 6548  ωcom 7870  cen 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-om 7871  df-en 8965
This theorem is referenced by:  findcard2OLD  9309
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