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Mirrors > Home > MPE Home > Th. List > alephfplem4 | Structured version Visualization version GIF version |
Description: Lemma for alephfp 10105. (Contributed by NM, 5-Nov-2004.) |
Ref | Expression |
---|---|
alephfplem.1 | β’ π» = (rec(β΅, Ο) βΎ Ο) |
Ref | Expression |
---|---|
alephfplem4 | β’ βͺ (π» β Ο) β ran β΅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 8437 | . . . . 5 β’ (rec(β΅, Ο) βΎ Ο) Fn Ο | |
2 | alephfplem.1 | . . . . . 6 β’ π» = (rec(β΅, Ο) βΎ Ο) | |
3 | 2 | fneq1i 6646 | . . . . 5 β’ (π» Fn Ο β (rec(β΅, Ο) βΎ Ο) Fn Ο) |
4 | 1, 3 | mpbir 230 | . . . 4 β’ π» Fn Ο |
5 | 2 | alephfplem3 10103 | . . . . 5 β’ (π§ β Ο β (π»βπ§) β ran β΅) |
6 | 5 | rgen 3063 | . . . 4 β’ βπ§ β Ο (π»βπ§) β ran β΅ |
7 | ffnfv 7120 | . . . 4 β’ (π»:ΟβΆran β΅ β (π» Fn Ο β§ βπ§ β Ο (π»βπ§) β ran β΅)) | |
8 | 4, 6, 7 | mpbir2an 709 | . . 3 β’ π»:ΟβΆran β΅ |
9 | ssun2 4173 | . . 3 β’ ran β΅ β (Ο βͺ ran β΅) | |
10 | fss 6734 | . . 3 β’ ((π»:ΟβΆran β΅ β§ ran β΅ β (Ο βͺ ran β΅)) β π»:ΟβΆ(Ο βͺ ran β΅)) | |
11 | 8, 9, 10 | mp2an 690 | . 2 β’ π»:ΟβΆ(Ο βͺ ran β΅) |
12 | peano1 7881 | . . 3 β’ β β Ο | |
13 | 2 | alephfplem1 10101 | . . 3 β’ (π»ββ ) β ran β΅ |
14 | fveq2 6891 | . . . . 5 β’ (π§ = β β (π»βπ§) = (π»ββ )) | |
15 | 14 | eleq1d 2818 | . . . 4 β’ (π§ = β β ((π»βπ§) β ran β΅ β (π»ββ ) β ran β΅)) |
16 | 15 | rspcev 3612 | . . 3 β’ ((β β Ο β§ (π»ββ ) β ran β΅) β βπ§ β Ο (π»βπ§) β ran β΅) |
17 | 12, 13, 16 | mp2an 690 | . 2 β’ βπ§ β Ο (π»βπ§) β ran β΅ |
18 | omex 9640 | . . 3 β’ Ο β V | |
19 | cardinfima 10094 | . . 3 β’ (Ο β V β ((π»:ΟβΆ(Ο βͺ ran β΅) β§ βπ§ β Ο (π»βπ§) β ran β΅) β βͺ (π» β Ο) β ran β΅)) | |
20 | 18, 19 | ax-mp 5 | . 2 β’ ((π»:ΟβΆ(Ο βͺ ran β΅) β§ βπ§ β Ο (π»βπ§) β ran β΅) β βͺ (π» β Ο) β ran β΅) |
21 | 11, 17, 20 | mp2an 690 | 1 β’ βͺ (π» β Ο) β ran β΅ |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 βwrex 3070 Vcvv 3474 βͺ cun 3946 β wss 3948 β c0 4322 βͺ cuni 4908 ran crn 5677 βΎ cres 5678 β cima 5679 Fn wfn 6538 βΆwf 6539 βcfv 6543 Οcom 7857 reccrdg 8411 β΅cale 9933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-har 9554 df-card 9936 df-aleph 9937 |
This theorem is referenced by: alephfp 10105 alephfp2 10106 |
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