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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 7604 | . . 3 ⊢ ∅ ∈ ω | |
2 | fvres 6692 | . . 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 8066 | . 2 ⊢ (𝐴 ∈ 𝐵 → (rec(𝐹, 𝐴)‘∅) = 𝐴) | |
5 | 3, 4 | syl5eq 2871 | 1 ⊢ (𝐴 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ ω)‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∅c0 4294 ↾ cres 5560 ‘cfv 6358 ωcom 7583 reccrdg 8048 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 |
This theorem is referenced by: unblem2 8774 dffi3 8898 inf0 9087 inf3lemb 9091 trcl 9173 alephfplem1 9533 infpssrlem1 9728 fin23lem34 9771 ituni0 9843 hsmexlem7 9848 axdclem2 9945 wunex2 10163 wuncval2 10172 peano5nni 11644 1nn 11652 om2uz0i 13318 om2uzrdg 13327 uzrdg0i 13330 trpredlem1 33070 trpredpred 33071 trpredmintr 33074 trpred0 33079 trpredrec 33081 neibastop2lem 33712 |
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