![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fr0g | Structured version Visualization version GIF version |
Description: The initial value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) |
Ref | Expression |
---|---|
fr0g | ⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7889 | . . 3 ⊢ ∅ ∈ ω | |
2 | fvres 6909 | . . 3 ⊢ (∅ ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ((rec(𝐹, 𝐴) ↾ ω)‘∅) = (rec(𝐹, 𝐴)‘∅) |
4 | rdg0g 8446 | . 2 ⊢ (𝐴 ∈ 𝐵 → (rec(𝐹, 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | eqtrid 2778 | 1 ⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∅c0 4322 ↾ cres 5674 ‘cfv 6543 ωcom 7865 reccrdg 8428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 |
This theorem is referenced by: unblem2 9320 dffi3 9464 inf0 9654 inf3lemb 9658 trcl 9761 alephfplem1 10137 infpssrlem1 10334 fin23lem34 10377 ituni0 10449 hsmexlem7 10454 axdclem2 10551 wunex2 10769 wuncval2 10778 peano5nni 12258 1nn 12266 om2uz0i 13958 om2uzrdg 13967 uzrdg0i 13970 noseq0 28258 noseqind 28260 om2noseq0 28264 om2noseqrdg 28272 noseqrdg0 28275 neibastop2lem 36082 |
Copyright terms: Public domain | W3C validator |