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Mirrors > Home > MPE Home > Th. List > tfrlem6 | Structured version Visualization version Unicode version |
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
Ref | Expression |
---|---|
tfrlem.1 |
Ref | Expression |
---|---|
tfrlem6 | recs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 5241 | . . 3 | |
2 | tfrlem.1 | . . . . 5 | |
3 | 2 | tfrlem4 7475 | . . . 4 |
4 | funrel 5905 | . . . 4 | |
5 | 3, 4 | syl 17 | . . 3 |
6 | 1, 5 | mprgbir 2927 | . 2 |
7 | 2 | recsfval 7477 | . . 3 recs |
8 | 7 | releqi 5202 | . 2 recs |
9 | 6, 8 | mpbir 221 | 1 recs |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 cab 2608 wral 2912 wrex 2913 cuni 4436 cres 5116 wrel 5119 con0 5723 wfun 5882 wfn 5883 cfv 5888 recscrecs 7467 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-wrecs 7407 df-recs 7468 |
This theorem is referenced by: tfrlem7 7479 tfrlem11 7484 tfrlem15 7488 tfrlem16 7489 |
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