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Theorem wfrlem8 7370
Description: Lemma for well-founded recursion. Compute the prececessor class for an 𝑅 minimal element of (𝐴 ∖ dom 𝐹). (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem6.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrlem8 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))

Proof of Theorem wfrlem8
StepHypRef Expression
1 wfrlem6.1 . . . . 5 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
21wfrdmss 7369 . . . 4 dom 𝐹𝐴
3 predpredss 5647 . . . 4 (dom 𝐹𝐴 → Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋))
42, 3ax-mp 5 . . 3 Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)
54biantru 526 . 2 (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ∧ Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)))
6 preddif 5666 . . . 4 Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋))
76eqeq1i 2626 . . 3 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
8 ssdif0 3918 . . 3 (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
97, 8bitr4i 267 . 2 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋))
10 eqss 3599 . 2 (Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ∧ Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)))
115, 9, 103bitr4i 292 1 (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  cdif 3553  wss 3556  c0 3893  dom cdm 5076  Predcpred 5640  wrecscwrecs 7354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-wrecs 7355
This theorem is referenced by:  wfrlem10  7372
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