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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 |
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Ref | Expression |
---|---|
tfrlem6 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reluni 4959 |
. . 3
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2 | tfrlem.1 |
. . . . 5
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3 | 2 | tfrlem4 7102 |
. . . 4
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4 | funrel 5602 |
. . . 4
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5 | 3, 4 | syl 17 |
. . 3
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6 | 1, 5 | mprgbir 2754 |
. 2
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7 | 2 | recsfval 7104 |
. . 3
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8 | 7 | releqi 4921 |
. 2
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9 | 6, 8 | mpbir 213 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-tr 4501 df-eprel 4748 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-iota 5549 df-fun 5587 df-fn 5588 df-fv 5593 df-wrecs 7033 df-recs 7095 |
This theorem is referenced by: tfrlem7 7106 tfrlem11 7111 tfrlem15 7115 tfrlem16 7116 |
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