Theorem wfrlem8OLD 8335

Description: Lemma for well-ordered recursion. Compute the prececessor class for an 𝑅 minimal element of (𝐴 ∖ dom 𝐹). Obsolete as of 18-Nov-2024. (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 8334	. . . 4 ⊢ dom 𝐹 ⊆ 𝐴
3		predpredss 6307	. . . 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 528	. 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 6330	. . . 4 ⊢ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋))
7	6	eqeq1i 2730	. . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
8		ssdif0 4359	. . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅)
9	7, 8	bitr4i 277	. 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋))
10		eqss 3988	. 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 302	1 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   ∖ cdif 3936   ⊆ wss 3939  ∅c0 4318  dom cdm 5672  Predcpred 6299  wrecscwrecs 8315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7419  df-2nd 7992  df-frecs 8285  df-wrecs 8316

This theorem is referenced by:  wfrlem10OLD  8337