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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 2905 | . 2 ⊢ Ⅎ𝑦𝐴 | |
| 2 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝐵 | |
| 3 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝐷 | |
| 4 | frsucmpt2.1 | . . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω) | |
| 5 | frsucmpt2.2 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
| 6 | 5 | cbvmptv 5255 | . . . . 5 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
| 7 | rdgeq1 8451 | . . . . 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 5993 | . . 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 8478 | 1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ↾ cres 5687 suc csuc 6386 ‘cfv 6561 ωcom 7887 reccrdg 8449 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 |
| This theorem is referenced by: unblem2 9329 unblem3 9330 inf0 9661 trcl 9768 hsmexlem8 10464 wunex2 10778 wuncval2 10787 peano5nni 12269 peano2nn 12278 om2uzsuci 13989 noseqp1 28297 noseqind 28298 om2noseqsuc 28303 dfnns2 28362 neibastop2lem 36361 |
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