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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 8385 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 6844 | . 2 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) |
3 | frfnom 8382 | . . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) Fn ω | |
4 | fndm 6606 | . . . . . 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 2830 | . . . 4 ⊢ (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) ↔ suc 𝐵 ∈ ω) |
7 | peano2b 7820 | . . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω) | |
8 | frsuc 8384 | . . . . . . . 8 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))) | |
9 | 1 | fveq1i 6844 | . . . . . . . . 9 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵) |
10 | 9 | fveq2i 6846 | . . . . . . . 8 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)) |
11 | 8, 10 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
12 | nfmpt1 5214 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
13 | frsucmpt.1 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐴 | |
14 | 12, 13 | nfrdg 8361 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
15 | nfcv 2908 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥ω | |
16 | 14, 15 | nfres 5940 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
17 | 1, 16 | nfcxfr 2906 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 |
18 | frsucmpt.2 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐵 | |
19 | 17, 18 | nffv 6853 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
20 | frsucmpt.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐷 | |
21 | frsucmpt.5 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
22 | eqid 2737 | . . . . . . . 8 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
23 | 19, 20, 21, 22 | fvmptnf 6971 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅) |
24 | 11, 23 | sylan9eqr 2799 | . . . . . 6 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) |
25 | 24 | ex 414 | . . . . 5 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
26 | 7, 25 | biimtrrid 242 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
27 | 6, 26 | biimtrid 241 | . . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅)) |
28 | ndmfv 6878 | . . 3 ⊢ (¬ suc 𝐵 ∈ dom (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) | |
29 | 27, 28 | pm2.61d1 180 | . 2 ⊢ (¬ 𝐷 ∈ V → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ∅) |
30 | 2, 29 | eqtrid 2789 | 1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2888 Vcvv 3446 ∅c0 4283 ↦ cmpt 5189 dom cdm 5634 ↾ cres 5636 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 reccrdg 8356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 |
This theorem is referenced by: (None) |
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