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Theorem wfrlem10 7662
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 7660 . . 3 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
32biimpi 208 . 2 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4 predss 5904 . . . 4 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
54a1i 11 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹)
6 simpr 478 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7 eldifn 3930 . . . . . . . 8 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8 eleq1w 2860 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹𝑧 ∈ dom 𝐹))
98notbid 310 . . . . . . . 8 (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
107, 9syl5ibrcom 239 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
1110con2d 132 . . . . . 6 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
1211imp 396 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
131wfrdmcl 7661 . . . . . . . 8 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
1413adantl 474 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
15 ssel 3791 . . . . . . . . . 10 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) → 𝑧 ∈ dom 𝐹))
1615con3d 150 . . . . . . . . 9 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (¬ 𝑧 ∈ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
177, 16syl5com 31 . . . . . . . 8 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
1817adantr 473 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
1914, 18mpd 15 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
20 eldifi 3929 . . . . . . 7 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
21 elpredg 5911 . . . . . . . 8 ((𝑤 ∈ dom 𝐹𝑧𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2221ancoms 451 . . . . . . 7 ((𝑧𝐴𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2320, 22sylan 576 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2419, 23mtbid 316 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
251wfrdmss 7659 . . . . . . 7 dom 𝐹𝐴
2625sseli 3793 . . . . . 6 (𝑤 ∈ dom 𝐹𝑤𝐴)
27 wfrlem10.1 . . . . . . . 8 𝑅 We 𝐴
28 weso 5302 . . . . . . . 8 (𝑅 We 𝐴𝑅 Or 𝐴)
2927, 28ax-mp 5 . . . . . . 7 𝑅 Or 𝐴
30 solin 5255 . . . . . . 7 ((𝑅 Or 𝐴 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3129, 30mpan 682 . . . . . 6 ((𝑤𝐴𝑧𝐴) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3226, 20, 31syl2anr 591 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3312, 24, 32ecase23d 1598 . . . 4 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
34 vex 3387 . . . . 5 𝑧 ∈ V
35 vex 3387 . . . . . 6 𝑤 ∈ V
3635elpred 5910 . . . . 5 (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧)))
3734, 36ax-mp 5 . . . 4 (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧))
386, 33, 37sylanbrc 579 . . 3 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
395, 38eqelssd 3817 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
403, 39sylan9eqr 2854 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 198  wa 385  w3o 1107   = wceq 1653  wcel 2157  Vcvv 3384  cdif 3765  wss 3768  c0 4114   class class class wbr 4842   Or wor 5231   We wwe 5269  dom cdm 5311  Predcpred 5896  wrecscwrecs 7643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-so 5233  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5897  df-iota 6063  df-fun 6102  df-fn 6103  df-fv 6108  df-wrecs 7644
This theorem is referenced by:  wfrlem15  7667
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