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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 7645 | . . 3 ⊢ ∅ ∈ ω | |
2 | fvres 6714 | . . 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 8141 | . 2 ⊢ (𝐴 ∈ 𝐵 → (rec(𝐹, 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | syl5eq 2783 | 1 ⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∅c0 4223 ↾ cres 5538 ‘cfv 6358 ωcom 7622 reccrdg 8123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 |
This theorem is referenced by: unblem2 8902 dffi3 9025 inf0 9214 inf3lemb 9218 trpredlem1 9310 trpredpred 9311 trpredmintr 9314 trpred0 9315 trpredrec 9320 trcl 9322 alephfplem1 9683 infpssrlem1 9882 fin23lem34 9925 ituni0 9997 hsmexlem7 10002 axdclem2 10099 wunex2 10317 wuncval2 10326 peano5nni 11798 1nn 11806 om2uz0i 13485 om2uzrdg 13494 uzrdg0i 13497 neibastop2lem 34235 |
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