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Mirrors > Home > MPE Home > Th. List > frsucmptn | Structured version Visualization version GIF version |
Description: The value of the finite recursive definition generator at a successor (special case where the characteristic function is a mapping abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with frsucmpt 8067 to help eliminate redundant sethood antecedents. (Contributed by Scott Fenton, 19-Feb-2011.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
frsucmpt.1 | ⊢ Ⅎ𝑥𝐴 |
frsucmpt.2 | ⊢ Ⅎ𝑥𝐵 |
frsucmpt.3 | ⊢ Ⅎ𝑥𝐷 |
frsucmpt.4 | ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
frsucmpt.5 | ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
frsucmptn | ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frsucmpt.4 | . . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
2 | 1 | fveq1i 6666 | . 2 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) |
3 | frfnom 8064 | . . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω | |
4 | fndm 6450 | . . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω → dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) = ω |
6 | 5 | eleq2i 2904 | . . . 4 ⊢ (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω) |
7 | peano2b 7590 | . . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω) | |
8 | frsuc 8066 | . . . . . . . 8 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))) | |
9 | 1 | fveq1i 6666 | . . . . . . . . 9 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵) |
10 | 9 | fveq2i 6668 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)) |
11 | 8, 10 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
12 | nfmpt1 5157 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
13 | frsucmpt.1 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴 | |
14 | 12, 13 | nfrdg 8044 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
15 | nfcv 2977 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfres 5850 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
17 | 1, 16 | nfcxfr 2975 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 |
18 | frsucmpt.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐵 | |
19 | 17, 18 | nffv 6675 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
20 | frsucmpt.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐷 | |
21 | frsucmpt.5 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
22 | eqid 2821 | . . . . . . . 8 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
23 | 19, 20, 21, 22 | fvmptnf 6785 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅) |
24 | 11, 23 | sylan9eqr 2878 | . . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) |
25 | 24 | ex 415 | . . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
26 | 7, 25 | syl5bir 245 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
27 | 6, 26 | syl5bi 244 | . . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
28 | ndmfv 6695 | . . 3 ⊢ (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) | |
29 | 27, 28 | pm2.61d1 182 | . 2 ⊢ (¬ 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) |
30 | 2, 29 | syl5eq 2868 | 1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3495 ∅c0 4291 ↦ cmpt 5139 dom cdm 5550 ↾ cres 5552 suc csuc 6188 Fn wfn 6345 ‘cfv 6350 ωcom 7574 reccrdg 8039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 |
This theorem is referenced by: trpredlem1 33061 |
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