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Theorem wfrlem10 7958
Description: Lemma for well-founded recursion. When 𝑧 is an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), then its predecessor class is equal to dom 𝐹. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem10.1 𝑅 We 𝐴
wfrlem10.2 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrlem10 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
Distinct variable group:   𝑧,𝐴
Allowed substitution hints:   𝑅(𝑧)   𝐹(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem10
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wfrlem10.2 . . . 4 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
21wfrlem8 7956 . . 3 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
32biimpi 217 . 2 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4 predss 6152 . . . 4 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
54a1i 11 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹)
6 simpr 485 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7 eldifn 4107 . . . . . . . 8 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8 eleq1w 2899 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹𝑧 ∈ dom 𝐹))
98notbid 319 . . . . . . . 8 (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
107, 9syl5ibrcom 248 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
1110con2d 136 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
1211imp 407 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
13 ssel 3964 . . . . . . . . 9 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) → 𝑧 ∈ dom 𝐹))
1413con3d 155 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (¬ 𝑧 ∈ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
157, 14syl5com 31 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
161wfrdmcl 7957 . . . . . . 7 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
1715, 16impel 506 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
18 eldifi 4106 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
19 elpredg 6159 . . . . . . . 8 ((𝑤 ∈ dom 𝐹𝑧𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2019ancoms 459 . . . . . . 7 ((𝑧𝐴𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2118, 20sylan 580 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2217, 21mtbid 325 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
231wfrdmss 7955 . . . . . . 7 dom 𝐹𝐴
2423sseli 3966 . . . . . 6 (𝑤 ∈ dom 𝐹𝑤𝐴)
25 wfrlem10.1 . . . . . . . 8 𝑅 We 𝐴
26 weso 5544 . . . . . . . 8 (𝑅 We 𝐴𝑅 Or 𝐴)
2725, 26ax-mp 5 . . . . . . 7 𝑅 Or 𝐴
28 solin 5496 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
2927, 28mpan 686 . . . . . 6 ((𝑤𝐴𝑧𝐴) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3024, 18, 29syl2anr 596 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3112, 22, 30ecase23d 1466 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
32 vex 3502 . . . . . 6 𝑤 ∈ V
3332elpred 6158 . . . . 5 (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧)))
3433elv 3504 . . . 4 (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧))
356, 31, 34sylanbrc 583 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
365, 35eqelssd 3991 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
373, 36sylan9eqr 2882 1 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3o 1080   = wceq 1530  wcel 2107  Vcvv 3499  cdif 3936  wss 3939  c0 4294   class class class wbr 5062   Or wor 5471   We wwe 5511  dom cdm 5553  Predcpred 6144  wrecscwrecs 7940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-so 5473  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359  df-wrecs 7941
This theorem is referenced by:  wfrlem15  7963
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