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Mirrors > Home > MPE Home > Th. List > wfrlem13 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. From here through wfrlem16 7972, we aim to prove that dom 𝐹 = 𝐴. We do this by supposing that there is an element 𝑧 of 𝐴 that is not in dom 𝐹. We then define 𝐶 by extending dom 𝐹 with the appropriate value at 𝑧. We then show that 𝑧 cannot be an 𝑅 minimal element of (𝐴 ∖ dom 𝐹), meaning that (𝐴 ∖ dom 𝐹) must be empty, so dom 𝐹 = 𝐴. Here, we show that 𝐶 is a function extending the domain of 𝐹 by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfrlem13.1 | ⊢ 𝑅 We 𝐴 |
wfrlem13.2 | ⊢ 𝑅 Se 𝐴 |
wfrlem13.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
wfrlem13.4 | ⊢ 𝐶 = (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
Ref | Expression |
---|---|
wfrlem13 | ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfrlem13.1 | . . . . 5 ⊢ 𝑅 We 𝐴 | |
2 | wfrlem13.2 | . . . . 5 ⊢ 𝑅 Se 𝐴 | |
3 | wfrlem13.3 | . . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | 1, 2, 3 | wfrfun 7967 | . . . 4 ⊢ Fun 𝐹 |
5 | vex 3499 | . . . . 5 ⊢ 𝑧 ∈ V | |
6 | fvex 6685 | . . . . 5 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
7 | 5, 6 | funsn 6409 | . . . 4 ⊢ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} |
8 | 4, 7 | pm3.2i 473 | . . 3 ⊢ (Fun 𝐹 ∧ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
9 | 6 | dmsnop 6075 | . . . . 5 ⊢ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} = {𝑧} |
10 | 9 | ineq2i 4188 | . . . 4 ⊢ (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∩ {𝑧}) |
11 | eldifn 4106 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
12 | disjsn 4649 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹) | |
13 | 11, 12 | sylibr 236 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅) |
14 | 10, 13 | syl5eq 2870 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = ∅) |
15 | funun 6402 | . . 3 ⊢ (((Fun 𝐹 ∧ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∧ (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = ∅) → Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉})) | |
16 | 8, 14, 15 | sylancr 589 | . 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉})) |
17 | dmun 5781 | . . 3 ⊢ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
18 | 9 | uneq2i 4138 | . . 3 ⊢ (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}) |
19 | 17, 18 | eqtri 2846 | . 2 ⊢ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}) |
20 | wfrlem13.4 | . . . 4 ⊢ 𝐶 = (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
21 | 20 | fneq1i 6452 | . . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn (dom 𝐹 ∪ {𝑧})) |
22 | df-fn 6360 | . . 3 ⊢ ((𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∧ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}))) | |
23 | 21, 22 | bitri 277 | . 2 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∧ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}))) |
24 | 16, 19, 23 | sylanblrc 592 | 1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 〈cop 4575 Se wse 5514 We wwe 5515 dom cdm 5557 ↾ cres 5559 Predcpred 6149 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 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 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 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-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 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: wfrlem14 7970 wfrlem15 7971 |
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