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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 7317, 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 | . . . . . 6 ⊢ 𝑅 We 𝐴 | |
2 | wfrlem13.2 | . . . . . 6 ⊢ 𝑅 Se 𝐴 | |
3 | wfrlem13.3 | . . . . . 6 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | 1, 2, 3 | wfrfun 7312 | . . . . 5 ⊢ Fun 𝐹 |
5 | vex 3176 | . . . . . 6 ⊢ 𝑧 ∈ V | |
6 | fvex 6113 | . . . . . 6 ⊢ (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
7 | 5, 6 | funsn 5853 | . . . . 5 ⊢ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} |
8 | 4, 7 | pm3.2i 470 | . . . 4 ⊢ (Fun 𝐹 ∧ Fun {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
9 | 6 | dmsnop 5527 | . . . . . 6 ⊢ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} = {𝑧} |
10 | 9 | ineq2i 3773 | . . . . 5 ⊢ (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∩ {𝑧}) |
11 | eldifn 3695 | . . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
12 | disjsn 4192 | . . . . . 6 ⊢ ((dom 𝐹 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ dom 𝐹) | |
13 | 11, 12 | sylibr 223 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ {𝑧}) = ∅) |
14 | 10, 13 | syl5eq 2656 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∩ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = ∅) |
15 | funun 5846 | . . . 4 ⊢ (((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 694 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉})) |
17 | dmun 5253 | . . . 4 ⊢ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
18 | 9 | uneq2i 3726 | . . . 4 ⊢ (dom 𝐹 ∪ dom {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}) |
19 | 17, 18 | eqtri 2632 | . . 3 ⊢ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}) |
20 | 16, 19 | jctir 559 | . 2 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (Fun (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∧ dom (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (dom 𝐹 ∪ {𝑧}))) |
21 | wfrlem13.4 | . . . 4 ⊢ 𝐶 = (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
22 | 21 | fneq1i 5899 | . . 3 ⊢ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ↔ (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn (dom 𝐹 ∪ {𝑧})) |
23 | df-fn 5807 | . . 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}))) | |
24 | 22, 23 | bitri 263 | . 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}))) |
25 | 20, 24 | sylibr 223 | 1 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 〈cop 4131 Se wse 4995 We wwe 4996 dom cdm 5038 ↾ cres 5040 Predcpred 5596 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 wrecscwrecs 7293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-wrecs 7294 |
This theorem is referenced by: wfrlem14 7315 wfrlem15 7316 |
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