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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 2932 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2932 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2932 | . 2 ⊢ Ⅎ𝑦𝐷 | |
4 | frsucmpt2.1 | . . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
5 | frsucmpt2.2 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
6 | 5 | cbvmptv 5028 | . . . . 5 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
7 | rdgeq1 7851 | . . . . 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 5691 | . . 3 ⊢ (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
10 | 4, 9 | eqtr4i 2805 | . 2 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) |
11 | frsucmpt2.3 | . 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) | |
12 | 1, 2, 3, 10, 11 | frsucmpt 7877 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ↦ cmpt 5008 ↾ cres 5409 suc csuc 6031 ‘cfv 6188 ωcom 7396 reccrdg 7849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 |
This theorem is referenced by: unblem2 8566 unblem3 8567 inf0 8878 trcl 8964 hsmexlem8 9644 wunex2 9958 wuncval2 9967 peano5nni 11442 peano2nn 11453 om2uzsuci 13131 neibastop2lem 33235 |
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