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Mirrors > Home > ILE Home > Th. List > tfrlem5 | Unicode version |
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
tfrlem.1 |
Ref | Expression |
---|---|
tfrlem5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem.1 | . . 3 | |
2 | vex 2663 | . . 3 | |
3 | 1, 2 | tfrlem3a 6175 | . 2 |
4 | vex 2663 | . . 3 | |
5 | 1, 4 | tfrlem3a 6175 | . 2 |
6 | reeanv 2577 | . . 3 | |
7 | fveq2 5389 | . . . . . . . . 9 | |
8 | fveq2 5389 | . . . . . . . . 9 | |
9 | 7, 8 | eqeq12d 2132 | . . . . . . . 8 |
10 | onin 4278 | . . . . . . . . . 10 | |
11 | 10 | 3ad2ant1 987 | . . . . . . . . 9 |
12 | simp2ll 1033 | . . . . . . . . . . 11 | |
13 | fnfun 5190 | . . . . . . . . . . 11 | |
14 | 12, 13 | syl 14 | . . . . . . . . . 10 |
15 | inss1 3266 | . . . . . . . . . . 11 | |
16 | fndm 5192 | . . . . . . . . . . . 12 | |
17 | 12, 16 | syl 14 | . . . . . . . . . . 11 |
18 | 15, 17 | sseqtrrid 3118 | . . . . . . . . . 10 |
19 | 14, 18 | jca 304 | . . . . . . . . 9 |
20 | simp2rl 1035 | . . . . . . . . . . 11 | |
21 | fnfun 5190 | . . . . . . . . . . 11 | |
22 | 20, 21 | syl 14 | . . . . . . . . . 10 |
23 | inss2 3267 | . . . . . . . . . . 11 | |
24 | fndm 5192 | . . . . . . . . . . . 12 | |
25 | 20, 24 | syl 14 | . . . . . . . . . . 11 |
26 | 23, 25 | sseqtrrid 3118 | . . . . . . . . . 10 |
27 | 22, 26 | jca 304 | . . . . . . . . 9 |
28 | simp2lr 1034 | . . . . . . . . . 10 | |
29 | ssralv 3131 | . . . . . . . . . 10 | |
30 | 15, 28, 29 | mpsyl 65 | . . . . . . . . 9 |
31 | simp2rr 1036 | . . . . . . . . . 10 | |
32 | ssralv 3131 | . . . . . . . . . 10 | |
33 | 23, 31, 32 | mpsyl 65 | . . . . . . . . 9 |
34 | 11, 19, 27, 30, 33 | tfrlem1 6173 | . . . . . . . 8 |
35 | simp3l 994 | . . . . . . . . . 10 | |
36 | fnbr 5195 | . . . . . . . . . 10 | |
37 | 12, 35, 36 | syl2anc 408 | . . . . . . . . 9 |
38 | simp3r 995 | . . . . . . . . . 10 | |
39 | fnbr 5195 | . . . . . . . . . 10 | |
40 | 20, 38, 39 | syl2anc 408 | . . . . . . . . 9 |
41 | elin 3229 | . . . . . . . . 9 | |
42 | 37, 40, 41 | sylanbrc 413 | . . . . . . . 8 |
43 | 9, 34, 42 | rspcdva 2768 | . . . . . . 7 |
44 | funbrfv 5428 | . . . . . . . 8 | |
45 | 14, 35, 44 | sylc 62 | . . . . . . 7 |
46 | funbrfv 5428 | . . . . . . . 8 | |
47 | 22, 38, 46 | sylc 62 | . . . . . . 7 |
48 | 43, 45, 47 | 3eqtr3d 2158 | . . . . . 6 |
49 | 48 | 3exp 1165 | . . . . 5 |
50 | 49 | rexlimdva 2526 | . . . 4 |
51 | 50 | rexlimiv 2520 | . . 3 |
52 | 6, 51 | sylbir 134 | . 2 |
53 | 3, 5, 52 | syl2anb 289 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 cin 3040 wss 3041 class class class wbr 3899 con0 4255 cdm 4509 cres 4511 wfun 5087 wfn 5088 cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: tfrlem7 6182 tfrexlem 6199 |
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