Theorem wfrlem8OLD 8355

Description: Obsolete version as of 18-Nov-2024. Lemma for well-ordered recursion. Compute the predecessor class for an 𝑅 minimal element of (𝐴 ∖ dom 𝐹). (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

Ref	Expression
wfrlem6OLD.1	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfrlem8OLD	⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))

Proof of Theorem wfrlem8OLD

Step	Hyp	Ref	Expression
1		wfrlem6OLD.1	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2	1	wfrdmssOLD 8354	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
3		predpredss 6330	. . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋))
4	2, 3	ax-mp 5	. . 3 ⊢ Pred(𝑅, dom 𝐹, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)
5	4	biantru 529	. 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 6352	. . . 4 ⊢ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋))
7	6	eqeq1i 2740	. . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
8		ssdif0 4372	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
9	7, 8	bitr4i 278	. 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋))
10		eqss 4011	. 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)))
11	5, 9, 10	3bitr4i 303	1 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  dom cdm 5689  Predcpred 6322  wrecscwrecs 8335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336

This theorem is referenced by:  wfrlem10OLD  8357