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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 2895 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2895 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2895 | . 2 ⊢ Ⅎ𝑦𝐷 | |
4 | frsucmpt2.1 | . . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
5 | frsucmpt2.2 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
6 | 5 | cbvmptv 5252 | . . . . 5 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
7 | rdgeq1 8407 | . . . . 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 5968 | . . 3 ⊢ (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) |
10 | 4, 9 | eqtr4i 2755 | . 2 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) |
11 | frsucmpt2.3 | . 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) | |
12 | 1, 2, 3, 10, 11 | frsucmpt 8434 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ↦ cmpt 5222 ↾ cres 5669 suc csuc 6357 ‘cfv 6534 ωcom 7849 reccrdg 8405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 |
This theorem is referenced by: unblem2 9293 unblem3 9294 inf0 9613 trcl 9720 hsmexlem8 10416 wunex2 10730 wuncval2 10739 peano5nni 12213 peano2nn 12222 om2uzsuci 13911 noseqp1 28083 noseqind 28084 om2noseqsuc 28089 neibastop2lem 35736 |
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