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Mirrors > Home > MPE Home > Th. List > frsucmpt2 | 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), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
frsucmpt2.1 | ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
frsucmpt2.2 | ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) |
frsucmpt2.3 | ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) |
Ref | Expression |
---|---|
frsucmpt2 | ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐷 | |
4 | frsucmpt2.1 | . . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
5 | frsucmpt2.2 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
6 | 5 | cbvmptv 5158 | . . . . 5 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
7 | rdgeq1 8147 | . . . . 5 ⊢ ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) | |
8 | 6, 7 | ax-mp 5 | . . . 4 ⊢ rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
9 | 8 | reseq1i 5847 | . . 3 ⊢ (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
10 | 4, 9 | eqtr4i 2768 | . 2 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) |
11 | frsucmpt2.3 | . 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) | |
12 | 1, 2, 3, 10, 11 | frsucmpt 8173 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ↦ cmpt 5135 ↾ cres 5553 suc csuc 6215 ‘cfv 6380 ωcom 7644 reccrdg 8145 |
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 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 |
This theorem is referenced by: unblem2 8924 unblem3 8925 inf0 9236 hsmexlem8 10038 wuncval2 10361 peano5nni 11833 peano2nn 11842 om2uzsuci 13521 neibastop2lem 34286 |
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