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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 7236 | . . 3 ⊢ ∅ ∈ ω | |
2 | fvres 6350 | . . 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 7680 | . 2 ⊢ (𝐴 ∈ 𝐵 → (rec(𝐹, 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | syl5eq 2817 | 1 ⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∅c0 4063 ↾ cres 5252 ‘cfv 6030 ωcom 7216 reccrdg 7662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-om 7217 df-wrecs 7563 df-recs 7625 df-rdg 7663 |
This theorem is referenced by: unblem2 8373 dffi3 8497 inf0 8686 inf3lemb 8690 trcl 8772 alephfplem1 9131 infpssrlem1 9331 fin23lem34 9374 ituni0 9446 hsmexlem7 9451 axdclem2 9548 wunex2 9766 wuncval2 9775 peano5nni 11229 1nn 11237 om2uz0i 12954 om2uzrdg 12963 uzrdg0i 12966 trpredlem1 32063 trpredpred 32064 trpredmintr 32067 trpred0 32072 trpredrec 32074 neibastop2lem 32692 cnfin0 33576 cnfinltrel 33577 |
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