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| 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.) | 
| Ref | Expression | 
|---|---|
| wfrlem6OLD.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | 
| Ref | Expression | 
|---|---|
| wfrlem8OLD | ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | wfrlem6OLD.1 | . . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
| 2 | 1 | wfrdmssOLD 8355 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 | 
| 3 | predpredss 6328 | . . . 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 6350 | . . . 4 ⊢ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) | |
| 7 | 6 | eqeq1i 2742 | . . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅) | 
| 8 | ssdif0 4366 | . . 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 3999 | . 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)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 dom cdm 5685 Predcpred 6320 wrecscwrecs 8336 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 | 
| This theorem is referenced by: wfrlem10OLD 8358 | 
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