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Mirrors > Home > MPE Home > Th. List > wfrlem13 | Structured version Visualization version GIF version |
Description: Lemma for well-ordered recursion. From here through wfrlem16 8070, 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 8065 | . . . 4 ⊢ Fun 𝐹 |
5 | vex 3412 | . . . . 5 ⊢ 𝑧 ∈ V | |
6 | fvex 6730 | . . . . 5 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
7 | 5, 6 | funsn 6433 | . . . 4 ⊢ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} |
8 | 4, 7 | pm3.2i 474 | . . 3 ⊢ (Fun 𝐹 ∧ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
9 | 6 | dmsnop 6079 | . . . . 5 ⊢ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} = {𝑧} |
10 | 9 | ineq2i 4124 | . . . 4 ⊢ (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∩ {𝑧}) |
11 | eldifn 4042 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
12 | disjsn 4627 | . . . . 5 ⊢ ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹) | |
13 | 11, 12 | sylibr 237 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅) |
14 | 10, 13 | eqtrid 2789 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = ∅) |
15 | funun 6426 | . . 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 590 | . 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉})) |
17 | dmun 5779 | . . 3 ⊢ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
18 | 9 | uneq2i 4074 | . . 3 ⊢ (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}) |
19 | 17, 18 | eqtri 2765 | . 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 6476 | . . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn (dom 𝐹 ∪ {𝑧})) |
22 | df-fn 6383 | . . 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 278 | . 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 593 | 1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 ∪ cun 3864 ∩ cin 3865 ∅c0 4237 {csn 4541 〈cop 4547 Se wse 5507 We wwe 5508 dom cdm 5551 ↾ cres 5553 Predcpred 6159 Fun wfun 6374 Fn wfn 6375 ‘cfv 6380 wrecscwrecs 8046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-iota 6338 df-fun 6382 df-fn 6383 df-fv 6388 df-wrecs 8047 |
This theorem is referenced by: wfrlem14 8068 wfrlem15 8069 |
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