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Mirrors > Home > MPE Home > Th. List > wfrlem8 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wfrlem6.1 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfrlem8 | ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, dom 𝐹, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrlem6.1 | . . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
2 | 1 | wfrdmss 7963 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
3 | predpredss 6156 | . . . 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 532 | . 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 6175 | . . . 4 ⊢ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) | |
7 | 6 | eqeq1i 2828 | . . 3 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅) |
8 | ssdif0 4325 | . . 3 ⊢ (Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋) ↔ (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, dom 𝐹, 𝑋)) = ∅) | |
9 | 7, 8 | bitr4i 280 | . 2 ⊢ (Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑋) = ∅ ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, dom 𝐹, 𝑋)) |
10 | eqss 3984 | . 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 305 | 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 208 ∧ wa 398 = wceq 1537 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 dom cdm 5557 Predcpred 6149 wrecscwrecs 7948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-wrecs 7949 |
This theorem is referenced by: wfrlem10 7966 |
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