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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 8370 | . . 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 6832 | . . 3 ⊢ (𝐹‘suc 𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘suc 𝐵) |
| 4 | 2 | fveq1i 6832 | . . . 4 ⊢ (𝐹‘𝐵) = ((rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)‘𝐵) |
| 5 | 4 | fveq2i 6834 | . . 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 6844 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
| 8 | nfmpt1 5174 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
| 9 | frsucmpt.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
| 10 | 8, 9 | nfrdg 8347 | . . . . . . 7 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
| 11 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥ω | |
| 12 | 10, 11 | nfres 5940 | . . . . . 6 ⊢ Ⅎ𝑥(rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
| 13 | 2, 12 | nfcxfr 2901 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
| 14 | frsucmpt.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 15 | 13, 14 | nffv 6841 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
| 16 | frsucmpt.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
| 17 | frsucmpt.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
| 18 | eqid 2741 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
| 19 | 15, 16, 17, 18 | fvmptf 6961 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
| 20 | 7, 19 | mpan 697 | . 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 397 = wceq 1548 ∈ wcel 2121 Ⅎwnfc 2888 Vcvv 3433 ↦ cmpt 5156 ↾ cres 5623 suc csuc 6316 ‘cfv 6489 ωcom 7810 reccrdg 8342 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-om 7811 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 |
| This theorem is referenced by: frsucmpt2 8373 dffi3 9338 axdclem 10436 |
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