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Theorem wfrlem10 7288
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 7286 . . 3 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
32biimpi 204 . 2 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) = Pred(𝑅, dom 𝐹, 𝑧))
4 predss 5590 . . . 4 Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹
54a1i 11 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹)
6 simpr 475 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ dom 𝐹)
7 eldifn 3694 . . . . . . . . . 10 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹)
8 eleq1 2675 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤 ∈ dom 𝐹𝑧 ∈ dom 𝐹))
98notbid 306 . . . . . . . . . 10 (𝑤 = 𝑧 → (¬ 𝑤 ∈ dom 𝐹 ↔ ¬ 𝑧 ∈ dom 𝐹))
107, 9syl5ibrcom 235 . . . . . . . . 9 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ dom 𝐹))
1110con2d 127 . . . . . . . 8 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹 → ¬ 𝑤 = 𝑧))
1211imp 443 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑤 = 𝑧)
131wfrdmcl 7287 . . . . . . . . . 10 (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
1413adantl 480 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹)
15 ssel 3561 . . . . . . . . . . . 12 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) → 𝑧 ∈ dom 𝐹))
1615con3d 146 . . . . . . . . . . 11 (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → (¬ 𝑧 ∈ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
177, 16syl5com 31 . . . . . . . . . 10 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
1817adantr 479 . . . . . . . . 9 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹 → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤)))
1914, 18mpd 15 . . . . . . . 8 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧 ∈ Pred(𝑅, 𝐴, 𝑤))
20 eldifi 3693 . . . . . . . . 9 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧𝐴)
21 elpredg 5597 . . . . . . . . . 10 ((𝑤 ∈ dom 𝐹𝑧𝐴) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2221ancoms 467 . . . . . . . . 9 ((𝑧𝐴𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2320, 22sylan 486 . . . . . . . 8 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑧 ∈ Pred(𝑅, 𝐴, 𝑤) ↔ 𝑧𝑅𝑤))
2419, 23mtbid 312 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → ¬ 𝑧𝑅𝑤)
251wfrdmss 7285 . . . . . . . . 9 dom 𝐹𝐴
2625sseli 3563 . . . . . . . 8 (𝑤 ∈ dom 𝐹𝑤𝐴)
27 wfrlem10.1 . . . . . . . . . 10 𝑅 We 𝐴
28 weso 5019 . . . . . . . . . 10 (𝑅 We 𝐴𝑅 Or 𝐴)
2927, 28ax-mp 5 . . . . . . . . 9 𝑅 Or 𝐴
30 solin 4972 . . . . . . . . 9 ((𝑅 Or 𝐴 ∧ (𝑤𝐴𝑧𝐴)) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3129, 30mpan 701 . . . . . . . 8 ((𝑤𝐴𝑧𝐴) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3226, 20, 31syl2anr 493 . . . . . . 7 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → (𝑤𝑅𝑧𝑤 = 𝑧𝑧𝑅𝑤))
3312, 24, 32ecase23d 1427 . . . . . 6 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤𝑅𝑧)
34 vex 3175 . . . . . . 7 𝑧 ∈ V
35 vex 3175 . . . . . . . 8 𝑤 ∈ V
3635elpred 5596 . . . . . . 7 (𝑧 ∈ V → (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧)))
3734, 36ax-mp 5 . . . . . 6 (𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧) ↔ (𝑤 ∈ dom 𝐹𝑤𝑅𝑧))
386, 33, 37sylanbrc 694 . . . . 5 ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ dom 𝐹) → 𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧))
3938ex 448 . . . 4 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑤 ∈ dom 𝐹𝑤 ∈ Pred(𝑅, dom 𝐹, 𝑧)))
4039ssrdv 3573 . . 3 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → dom 𝐹 ⊆ Pred(𝑅, dom 𝐹, 𝑧))
415, 40eqssd 3584 . 2 (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, dom 𝐹, 𝑧) = dom 𝐹)
423, 41sylan9eqr 2665 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 194  wa 382  w3o 1029   = wceq 1474  wcel 1976  Vcvv 3172  cdif 3536  wss 3539  c0 3873   class class class wbr 4577   Or wor 4948   We wwe 4986  dom cdm 5028  Predcpred 5582  wrecscwrecs 7270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-so 4950  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-wrecs 7271
This theorem is referenced by:  wfrlem15  7293
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