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Mirrors > Home > MPE Home > Th. List > frsucmpt | Structured version Visualization version GIF version |
Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation). (Contributed by NM, 14-Sep-2003.) (Revised by Scott Fenton, 2-Nov-2011.) |
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 |
---|---|
frsucmpt | ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frsuc 7419 | . . 3 ⊢ (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵))) | |
2 | frsucmpt.4 | . . . 4 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
3 | 2 | fveq1i 6104 | . . 3 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) |
4 | 2 | fveq1i 6104 | . . . 4 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵) |
5 | 4 | fveq2i 6106 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)) |
6 | 1, 3, 5 | 3eqtr4g 2669 | . 2 ⊢ (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
7 | fvex 6113 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
8 | nfmpt1 4675 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
9 | frsucmpt.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
10 | 8, 9 | nfrdg 7397 | . . . . . . 7 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
11 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥ω | |
12 | 10, 11 | nfres 5319 | . . . . . 6 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
13 | 2, 12 | nfcxfr 2749 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
14 | frsucmpt.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
15 | 13, 14 | nffv 6110 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
16 | frsucmpt.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
17 | frsucmpt.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
18 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
19 | 15, 16, 17, 18 | fvmptf 6209 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
20 | 7, 19 | mpan 702 | . 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
21 | 6, 20 | sylan9eq 2664 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 ↦ cmpt 4643 ↾ cres 5040 suc csuc 5642 ‘cfv 5804 ωcom 6957 reccrdg 7392 |
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 ax-un 6847 |
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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 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-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 |
This theorem is referenced by: frsucmpt2 7422 dffi3 8220 axdclem 9224 trpredlem1 30971 trpredtr 30974 trpredmintr 30975 trpred0 30980 trpredrec 30982 |
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