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Mirrors > Home > MPE Home > Th. List > tfrlem6 | 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 5130 | . . 3 | |
2 | tfrlem.1 | . . . . 5 | |
3 | 2 | tfrlem4 7067 | . . . 4 |
4 | funrel 5610 | . . . 4 | |
5 | 3, 4 | syl 16 | . . 3 |
6 | 1, 5 | mprgbir 2821 | . 2 |
7 | 2 | recsfval 7069 | . . 3 |
8 | 7 | releqi 5091 | . 2 |
9 | 6, 8 | mpbir 209 | 1 |
Colors of variables: wff setvar class |
Syntax hints: /\ wa 369 = wceq 1395
e. wcel 1818 { cab 2442 A. wral 2807
E. wrex 2808 U. cuni 4249 con0 4883 |` cres 5006 Rel wrel 5009
Fun wfun 5587
Fn wfn 5588 ` cfv 5593 recs crecs 7060 |
This theorem is referenced by: tfrlem7 7071 tfrlem11 7076 tfrlem15 7080 tfrlem16 7081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-res 5016 df-iota 5556 df-fun 5595 df-fn 5596 df-fv 5601 df-recs 7061 |
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