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Theorem wfrlem10 7975
 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 7973 . . 3 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
32biimpi 219 . 2 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4 predss 6134 . . . 4 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
54a1i 11 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹)
6 simpr 489 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7 eldifn 4034 . . . . . . . 8 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8 eleq1w 2835 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹𝑧 ∈ dom 𝐹))
98notbid 322 . . . . . . . 8 (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
107, 9syl5ibrcom 250 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
1110con2d 136 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
1211imp 411 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
13 ssel 3886 . . . . . . . . 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 7974 . . . . . . 7 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
1715, 16impel 510 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
18 eldifi 4033 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
19 elpredg 6141 . . . . . . . 8 ((𝑤 ∈ dom 𝐹𝑧𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2019ancoms 463 . . . . . . 7 ((𝑧𝐴𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2118, 20sylan 584 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2217, 21mtbid 328 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
231wfrdmss 7972 . . . . . . 7 dom 𝐹𝐴
2423sseli 3889 . . . . . 6 (𝑤 ∈ dom 𝐹𝑤𝐴)
25 wfrlem10.1 . . . . . . . 8 𝑅 We 𝐴
26 weso 5516 . . . . . . . 8 (𝑅 We 𝐴𝑅 Or 𝐴)
2725, 26ax-mp 5 . . . . . . 7 𝑅 Or 𝐴
28 solin 5468 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
2927, 28mpan 690 . . . . . 6 ((𝑤𝐴𝑧𝐴) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3024, 18, 29syl2anr 600 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3112, 22, 30ecase23d 1471 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
32 vex 3414 . . . . . 6 𝑤 ∈ V
3332elpred 6140 . . . . 5 (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧)))
3433elv 3416 . . . 4 (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧))
356, 31, 34sylanbrc 587 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
365, 35eqelssd 3914 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
373, 36sylan9eqr 2816 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 209   ∧ wa 400   ∨ w3o 1084   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4226   class class class wbr 5033   Or wor 5443   We wwe 5483  dom cdm 5525  Predcpred 6126  wrecscwrecs 7957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-so 5445  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-wrecs 7958 This theorem is referenced by:  wfrlem15  7980
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