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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 2560 | . . 3 | |
3 | 1, 2 | tfrlem3a 5925 | . 2 |
4 | vex 2560 | . . 3 | |
5 | 1, 4 | tfrlem3a 5925 | . 2 |
6 | reeanv 2479 | . . 3 | |
7 | simp2ll 971 | . . . . . . . . . 10 | |
8 | simp3l 932 | . . . . . . . . . 10 | |
9 | fnbr 5001 | . . . . . . . . . 10 | |
10 | 7, 8, 9 | syl2anc 391 | . . . . . . . . 9 |
11 | simp2rl 973 | . . . . . . . . . 10 | |
12 | simp3r 933 | . . . . . . . . . 10 | |
13 | fnbr 5001 | . . . . . . . . . 10 | |
14 | 11, 12, 13 | syl2anc 391 | . . . . . . . . 9 |
15 | elin 3126 | . . . . . . . . 9 | |
16 | 10, 14, 15 | sylanbrc 394 | . . . . . . . 8 |
17 | onin 4123 | . . . . . . . . . 10 | |
18 | 17 | 3ad2ant1 925 | . . . . . . . . 9 |
19 | fnfun 4996 | . . . . . . . . . . 11 | |
20 | 7, 19 | syl 14 | . . . . . . . . . 10 |
21 | inss1 3157 | . . . . . . . . . . 11 | |
22 | fndm 4998 | . . . . . . . . . . . 12 | |
23 | 7, 22 | syl 14 | . . . . . . . . . . 11 |
24 | 21, 23 | syl5sseqr 2994 | . . . . . . . . . 10 |
25 | 20, 24 | jca 290 | . . . . . . . . 9 |
26 | fnfun 4996 | . . . . . . . . . . 11 | |
27 | 11, 26 | syl 14 | . . . . . . . . . 10 |
28 | inss2 3158 | . . . . . . . . . . 11 | |
29 | fndm 4998 | . . . . . . . . . . . 12 | |
30 | 11, 29 | syl 14 | . . . . . . . . . . 11 |
31 | 28, 30 | syl5sseqr 2994 | . . . . . . . . . 10 |
32 | 27, 31 | jca 290 | . . . . . . . . 9 |
33 | simp2lr 972 | . . . . . . . . . 10 | |
34 | ssralv 3004 | . . . . . . . . . 10 | |
35 | 21, 33, 34 | mpsyl 59 | . . . . . . . . 9 |
36 | simp2rr 974 | . . . . . . . . . 10 | |
37 | ssralv 3004 | . . . . . . . . . 10 | |
38 | 28, 36, 37 | mpsyl 59 | . . . . . . . . 9 |
39 | 18, 25, 32, 35, 38 | tfrlem1 5923 | . . . . . . . 8 |
40 | fveq2 5178 | . . . . . . . . . 10 | |
41 | fveq2 5178 | . . . . . . . . . 10 | |
42 | 40, 41 | eqeq12d 2054 | . . . . . . . . 9 |
43 | 42 | rspcv 2652 | . . . . . . . 8 |
44 | 16, 39, 43 | sylc 56 | . . . . . . 7 |
45 | funbrfv 5212 | . . . . . . . 8 | |
46 | 20, 8, 45 | sylc 56 | . . . . . . 7 |
47 | funbrfv 5212 | . . . . . . . 8 | |
48 | 27, 12, 47 | sylc 56 | . . . . . . 7 |
49 | 44, 46, 48 | 3eqtr3d 2080 | . . . . . 6 |
50 | 49 | 3exp 1103 | . . . . 5 |
51 | 50 | rexlimdva 2433 | . . . 4 |
52 | 51 | rexlimiv 2427 | . . 3 |
53 | 6, 52 | sylbir 125 | . 2 |
54 | 3, 5, 53 | syl2anb 275 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wcel 1393 cab 2026 wral 2306 wrex 2307 cin 2916 wss 2917 class class class wbr 3764 con0 4100 cdm 4345 cres 4347 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: tfrlem7 5933 tfrexlem 5948 |
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