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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec1lem4 | Structured version Visualization version Unicode version |
Description: Lemma for setrec1 42438. If is recursively generated by , then
so is .
In the proof of setrec1 42438, the following is substituted for this theorem's : Therefore, we cannot declare to be a distinct variable from , since we need it to appear as a bound variable in . This theorem can be proven without the hypothesis , but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1762, making the antecedent of each line something more complicated than . The proof of setrec1lem2 42435 could similarly be made easier to read by adding the hypothesis , but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
setrec1lem4.1 | |
setrec1lem4.2 | |
setrec1lem4.3 | |
setrec1lem4.4 | |
setrec1lem4.5 |
Ref | Expression |
---|---|
setrec1lem4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setrec1lem4.1 | . . 3 | |
2 | id 22 | . . . . . . . 8 | |
3 | ssun1 3776 | . . . . . . . 8 | |
4 | 2, 3 | syl6ss 3615 | . . . . . . 7 |
5 | 4 | imim1i 63 | . . . . . 6 |
6 | 5 | alimi 1739 | . . . . 5 |
7 | setrec1lem4.5 | . . . . . . . 8 | |
8 | setrec1lem4.2 | . . . . . . . . 9 | |
9 | 8, 7 | setrec1lem1 42434 | . . . . . . . 8 |
10 | 7, 9 | mpbid 222 | . . . . . . 7 |
11 | sp 2053 | . . . . . . 7 | |
12 | 10, 11 | syl 17 | . . . . . 6 |
13 | setrec1lem4.4 | . . . . . . . . 9 | |
14 | sstr2 3610 | . . . . . . . . 9 | |
15 | 13, 14 | syl 17 | . . . . . . . 8 |
16 | 12, 15 | syld 47 | . . . . . . 7 |
17 | setrec1lem4.3 | . . . . . . . . 9 | |
18 | sseq1 3626 | . . . . . . . . . 10 | |
19 | sseq1 3626 | . . . . . . . . . . 11 | |
20 | fveq2 6191 | . . . . . . . . . . . 12 | |
21 | 20 | sseq1d 3632 | . . . . . . . . . . 11 |
22 | 19, 21 | imbi12d 334 | . . . . . . . . . 10 |
23 | 18, 22 | imbi12d 334 | . . . . . . . . 9 |
24 | 17, 23 | spcdvw 42426 | . . . . . . . 8 |
25 | 13, 24 | mpid 44 | . . . . . . 7 |
26 | 16, 25 | mpdd 43 | . . . . . 6 |
27 | 12, 26 | jcad 555 | . . . . 5 |
28 | 6, 27 | syl5 34 | . . . 4 |
29 | unss 3787 | . . . 4 | |
30 | 28, 29 | syl6ib 241 | . . 3 |
31 | 1, 30 | alrimi 2082 | . 2 |
32 | fvex 6201 | . . . 4 | |
33 | unexg 6959 | . . . 4 | |
34 | 7, 32, 33 | sylancl 694 | . . 3 |
35 | 8, 34 | setrec1lem1 42434 | . 2 |
36 | 31, 35 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wceq 1483 wnf 1708 wcel 1990 cab 2608 cvv 3200 cun 3572 wss 3574 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 |
This theorem is referenced by: setrec1 42438 |
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