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Mirrors > Home > MPE Home > Th. List > rdgeq1 | Structured version Visualization version Unicode version |
Description: Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
rdgeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6190 | . . . . . 6 | |
2 | 1 | ifeq2d 4105 | . . . . 5 |
3 | 2 | ifeq2d 4105 | . . . 4 |
4 | 3 | mpteq2dv 4745 | . . 3 |
5 | recseq 7470 | . . 3 recs recs | |
6 | 4, 5 | syl 17 | . 2 recs recs |
7 | df-rdg 7506 | . 2 recs | |
8 | df-rdg 7506 | . 2 recs | |
9 | 6, 7, 8 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 cvv 3200 c0 3915 cif 4086 cuni 4436 cmpt 4729 cdm 5114 crn 5115 wlim 5724 cfv 5888 recscrecs 7467 crdg 7505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 |
This theorem is referenced by: rdgeq12 7509 rdgsucmpt2 7526 frsucmpt2 7535 seqomlem0 7544 omv 7592 oev 7594 dffi3 8337 hsmex 9254 axdc 9343 seqeq2 12805 seqval 12812 trpredlem1 31727 trpredtr 31730 trpredmintr 31731 neibastop2 32356 dffinxpf 33222 finxpeq1 33223 |
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