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| Mirrors > Home > MPE Home > Th. List > rdgsuc | Structured version Visualization version GIF version | ||
| Description: The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| rdgsuc | ⊢ (𝐵 ∈ On → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rdgfnon 8459 | . . . 4 ⊢ rec(𝐹, 𝐴) Fn On | |
| 2 | 1 | fndmi 6671 | . . 3 ⊢ dom rec(𝐹, 𝐴) = On | 
| 3 | 2 | eleq2i 2832 | . 2 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) ↔ 𝐵 ∈ On) | 
| 4 | rdgsucg 8464 | . 2 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) | |
| 5 | 3, 4 | sylbir 235 | 1 ⊢ (𝐵 ∈ On → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 dom cdm 5684 Oncon0 6383 suc csuc 6385 ‘cfv 6560 reccrdg 8450 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 | 
| This theorem is referenced by: rdgsucmptf 8469 oasuc 8563 omsuc 8565 oesuc 8566 alephsuc 10109 ackbij2lem3 10281 constrsuc 33780 satfvsuc 35367 satf0suc 35382 sat1el2xp 35385 fmlasuc0 35390 rdgprc 35796 findreccl 36455 rdgsucuni 37371 rdgeqoa 37372 finxpreclem4 37396 finxpreclem6 37398 | 
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