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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 8478 | . . 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 6906 | . . 3 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) | 
| 4 | 2 | fveq1i 6906 | . . . 4 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵) | 
| 5 | 4 | fveq2i 6908 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵)) | 
| 6 | 1, 3, 5 | 3eqtr4g 2801 | . 2 ⊢ (𝐵 ∈ ω → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) | 
| 7 | fvex 6918 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
| 8 | nfmpt1 5249 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
| 9 | frsucmpt.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
| 10 | 8, 9 | nfrdg 8455 | . . . . . . 7 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | 
| 11 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥ω | |
| 12 | 10, 11 | nfres 5998 | . . . . . 6 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | 
| 13 | 2, 12 | nfcxfr 2902 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | 
| 14 | frsucmpt.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 15 | 13, 14 | nffv 6915 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) | 
| 16 | frsucmpt.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
| 17 | frsucmpt.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
| 18 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
| 19 | 15, 16, 17, 18 | fvmptf 7036 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) | 
| 20 | 7, 19 | mpan 690 | . 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) | 
| 21 | 6, 20 | sylan9eq 2796 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Ⅎwnfc 2889 Vcvv 3479 ↦ cmpt 5224 ↾ cres 5686 suc csuc 6385 ‘cfv 6560 ωcom 7888 reccrdg 8450 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 | 
| This theorem is referenced by: frsucmpt2 8481 dffi3 9472 axdclem 10560 | 
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