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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 2724 | . . 3 | |
3 | 1, 2 | tfrlem3a 6269 | . 2 |
4 | vex 2724 | . . 3 | |
5 | 1, 4 | tfrlem3a 6269 | . 2 |
6 | reeanv 2633 | . . 3 | |
7 | fveq2 5480 | . . . . . . . . 9 | |
8 | fveq2 5480 | . . . . . . . . 9 | |
9 | 7, 8 | eqeq12d 2179 | . . . . . . . 8 |
10 | onin 4358 | . . . . . . . . . 10 | |
11 | 10 | 3ad2ant1 1007 | . . . . . . . . 9 |
12 | simp2ll 1053 | . . . . . . . . . . 11 | |
13 | fnfun 5279 | . . . . . . . . . . 11 | |
14 | 12, 13 | syl 14 | . . . . . . . . . 10 |
15 | inss1 3337 | . . . . . . . . . . 11 | |
16 | fndm 5281 | . . . . . . . . . . . 12 | |
17 | 12, 16 | syl 14 | . . . . . . . . . . 11 |
18 | 15, 17 | sseqtrrid 3188 | . . . . . . . . . 10 |
19 | 14, 18 | jca 304 | . . . . . . . . 9 |
20 | simp2rl 1055 | . . . . . . . . . . 11 | |
21 | fnfun 5279 | . . . . . . . . . . 11 | |
22 | 20, 21 | syl 14 | . . . . . . . . . 10 |
23 | inss2 3338 | . . . . . . . . . . 11 | |
24 | fndm 5281 | . . . . . . . . . . . 12 | |
25 | 20, 24 | syl 14 | . . . . . . . . . . 11 |
26 | 23, 25 | sseqtrrid 3188 | . . . . . . . . . 10 |
27 | 22, 26 | jca 304 | . . . . . . . . 9 |
28 | simp2lr 1054 | . . . . . . . . . 10 | |
29 | ssralv 3201 | . . . . . . . . . 10 | |
30 | 15, 28, 29 | mpsyl 65 | . . . . . . . . 9 |
31 | simp2rr 1056 | . . . . . . . . . 10 | |
32 | ssralv 3201 | . . . . . . . . . 10 | |
33 | 23, 31, 32 | mpsyl 65 | . . . . . . . . 9 |
34 | 11, 19, 27, 30, 33 | tfrlem1 6267 | . . . . . . . 8 |
35 | simp3l 1014 | . . . . . . . . . 10 | |
36 | fnbr 5284 | . . . . . . . . . 10 | |
37 | 12, 35, 36 | syl2anc 409 | . . . . . . . . 9 |
38 | simp3r 1015 | . . . . . . . . . 10 | |
39 | fnbr 5284 | . . . . . . . . . 10 | |
40 | 20, 38, 39 | syl2anc 409 | . . . . . . . . 9 |
41 | elin 3300 | . . . . . . . . 9 | |
42 | 37, 40, 41 | sylanbrc 414 | . . . . . . . 8 |
43 | 9, 34, 42 | rspcdva 2830 | . . . . . . 7 |
44 | funbrfv 5519 | . . . . . . . 8 | |
45 | 14, 35, 44 | sylc 62 | . . . . . . 7 |
46 | funbrfv 5519 | . . . . . . . 8 | |
47 | 22, 38, 46 | sylc 62 | . . . . . . 7 |
48 | 43, 45, 47 | 3eqtr3d 2205 | . . . . . 6 |
49 | 48 | 3exp 1191 | . . . . 5 |
50 | 49 | rexlimdva 2581 | . . . 4 |
51 | 50 | rexlimiv 2575 | . . 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 967 wceq 1342 wcel 2135 cab 2150 wral 2442 wrex 2443 cin 3110 wss 3111 class class class wbr 3976 con0 4335 cdm 4598 cres 4600 wfun 5176 wfn 5177 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: tfrlem7 6276 tfrexlem 6293 |
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