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Theorem aomclem4 41387
Description: Lemma for dfac11 41392. Limit case. Patch together well-orderings constructed so far using fnwe2 41383 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
aomclem4.f 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
aomclem4.on (πœ‘ β†’ dom 𝑧 ∈ On)
aomclem4.su (πœ‘ β†’ dom 𝑧 = βˆͺ dom 𝑧)
aomclem4.we (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
Assertion
Ref Expression
aomclem4 (πœ‘ β†’ 𝐹 We (𝑅1β€˜dom 𝑧))
Distinct variable groups:   𝑧,π‘Ž,𝑏   πœ‘,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(𝑧)   𝐹(𝑧,π‘Ž,𝑏)

Proof of Theorem aomclem4
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 suceq 6384 . . 3 (𝑐 = (rankβ€˜π‘Ž) β†’ suc 𝑐 = suc (rankβ€˜π‘Ž))
21fveq2d 6847 . 2 (𝑐 = (rankβ€˜π‘Ž) β†’ (π‘§β€˜suc 𝑐) = (π‘§β€˜suc (rankβ€˜π‘Ž)))
3 aomclem4.f . 2 𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}
4 r1fnon 9704 . . . . . . . . . . . . . 14 𝑅1 Fn On
5 fnfun 6603 . . . . . . . . . . . . . 14 (𝑅1 Fn On β†’ Fun 𝑅1)
64, 5ax-mp 5 . . . . . . . . . . . . 13 Fun 𝑅1
74fndmi 6607 . . . . . . . . . . . . . 14 dom 𝑅1 = On
87eqimss2i 4004 . . . . . . . . . . . . 13 On βŠ† dom 𝑅1
96, 8pm3.2i 472 . . . . . . . . . . . 12 (Fun 𝑅1 ∧ On βŠ† dom 𝑅1)
10 aomclem4.on . . . . . . . . . . . 12 (πœ‘ β†’ dom 𝑧 ∈ On)
11 funfvima2 7182 . . . . . . . . . . . 12 ((Fun 𝑅1 ∧ On βŠ† dom 𝑅1) β†’ (dom 𝑧 ∈ On β†’ (𝑅1β€˜dom 𝑧) ∈ (𝑅1 β€œ On)))
129, 10, 11mpsyl 68 . . . . . . . . . . 11 (πœ‘ β†’ (𝑅1β€˜dom 𝑧) ∈ (𝑅1 β€œ On))
13 elssuni 4899 . . . . . . . . . . 11 ((𝑅1β€˜dom 𝑧) ∈ (𝑅1 β€œ On) β†’ (𝑅1β€˜dom 𝑧) βŠ† βˆͺ (𝑅1 β€œ On))
1412, 13syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝑅1β€˜dom 𝑧) βŠ† βˆͺ (𝑅1 β€œ On))
1514sselda 3945 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ (𝑅1β€˜dom 𝑧)) β†’ 𝑏 ∈ βˆͺ (𝑅1 β€œ On))
16 rankidb 9737 . . . . . . . . 9 (𝑏 ∈ βˆͺ (𝑅1 β€œ On) β†’ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘)))
1715, 16syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ (𝑅1β€˜dom 𝑧)) β†’ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘)))
18 suceq 6384 . . . . . . . . . 10 ((rankβ€˜π‘) = (rankβ€˜π‘Ž) β†’ suc (rankβ€˜π‘) = suc (rankβ€˜π‘Ž))
1918fveq2d 6847 . . . . . . . . 9 ((rankβ€˜π‘) = (rankβ€˜π‘Ž) β†’ (𝑅1β€˜suc (rankβ€˜π‘)) = (𝑅1β€˜suc (rankβ€˜π‘Ž)))
2019eleq2d 2824 . . . . . . . 8 ((rankβ€˜π‘) = (rankβ€˜π‘Ž) β†’ (𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘)) ↔ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘Ž))))
2117, 20syl5ibcom 244 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ (𝑅1β€˜dom 𝑧)) β†’ ((rankβ€˜π‘) = (rankβ€˜π‘Ž) β†’ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘Ž))))
2221expimpd 455 . . . . . 6 (πœ‘ β†’ ((𝑏 ∈ (𝑅1β€˜dom 𝑧) ∧ (rankβ€˜π‘) = (rankβ€˜π‘Ž)) β†’ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘Ž))))
2322ss2abdv 4021 . . . . 5 (πœ‘ β†’ {𝑏 ∣ (𝑏 ∈ (𝑅1β€˜dom 𝑧) ∧ (rankβ€˜π‘) = (rankβ€˜π‘Ž))} βŠ† {𝑏 ∣ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘Ž))})
24 df-rab 3409 . . . . 5 {𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)} = {𝑏 ∣ (𝑏 ∈ (𝑅1β€˜dom 𝑧) ∧ (rankβ€˜π‘) = (rankβ€˜π‘Ž))}
25 abid1 2875 . . . . 5 (𝑅1β€˜suc (rankβ€˜π‘Ž)) = {𝑏 ∣ 𝑏 ∈ (𝑅1β€˜suc (rankβ€˜π‘Ž))}
2623, 24, 253sstr4g 3990 . . . 4 (πœ‘ β†’ {𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)} βŠ† (𝑅1β€˜suc (rankβ€˜π‘Ž)))
2726adantr 482 . . 3 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ {𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)} βŠ† (𝑅1β€˜suc (rankβ€˜π‘Ž)))
28 fveq2 6843 . . . . 5 (𝑏 = suc (rankβ€˜π‘Ž) β†’ (π‘§β€˜π‘) = (π‘§β€˜suc (rankβ€˜π‘Ž)))
29 fveq2 6843 . . . . 5 (𝑏 = suc (rankβ€˜π‘Ž) β†’ (𝑅1β€˜π‘) = (𝑅1β€˜suc (rankβ€˜π‘Ž)))
3028, 29weeq12d 41370 . . . 4 (𝑏 = suc (rankβ€˜π‘Ž) β†’ ((π‘§β€˜π‘) We (𝑅1β€˜π‘) ↔ (π‘§β€˜suc (rankβ€˜π‘Ž)) We (𝑅1β€˜suc (rankβ€˜π‘Ž))))
31 aomclem4.we . . . . . 6 (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))
32 fveq2 6843 . . . . . . . 8 (π‘Ž = 𝑏 β†’ (π‘§β€˜π‘Ž) = (π‘§β€˜π‘))
33 fveq2 6843 . . . . . . . 8 (π‘Ž = 𝑏 β†’ (𝑅1β€˜π‘Ž) = (𝑅1β€˜π‘))
3432, 33weeq12d 41370 . . . . . . 7 (π‘Ž = 𝑏 β†’ ((π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž) ↔ (π‘§β€˜π‘) We (𝑅1β€˜π‘)))
3534cbvralvw 3226 . . . . . 6 (βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž) ↔ βˆ€π‘ ∈ dom 𝑧(π‘§β€˜π‘) We (𝑅1β€˜π‘))
3631, 35sylib 217 . . . . 5 (πœ‘ β†’ βˆ€π‘ ∈ dom 𝑧(π‘§β€˜π‘) We (𝑅1β€˜π‘))
3736adantr 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ βˆ€π‘ ∈ dom 𝑧(π‘§β€˜π‘) We (𝑅1β€˜π‘))
38 rankr1ai 9735 . . . . . 6 (π‘Ž ∈ (𝑅1β€˜dom 𝑧) β†’ (rankβ€˜π‘Ž) ∈ dom 𝑧)
3938adantl 483 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ (rankβ€˜π‘Ž) ∈ dom 𝑧)
40 eloni 6328 . . . . . . . 8 (dom 𝑧 ∈ On β†’ Ord dom 𝑧)
4110, 40syl 17 . . . . . . 7 (πœ‘ β†’ Ord dom 𝑧)
42 aomclem4.su . . . . . . 7 (πœ‘ β†’ dom 𝑧 = βˆͺ dom 𝑧)
43 limsuc2 41371 . . . . . . 7 ((Ord dom 𝑧 ∧ dom 𝑧 = βˆͺ dom 𝑧) β†’ ((rankβ€˜π‘Ž) ∈ dom 𝑧 ↔ suc (rankβ€˜π‘Ž) ∈ dom 𝑧))
4441, 42, 43syl2anc 585 . . . . . 6 (πœ‘ β†’ ((rankβ€˜π‘Ž) ∈ dom 𝑧 ↔ suc (rankβ€˜π‘Ž) ∈ dom 𝑧))
4544adantr 482 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ ((rankβ€˜π‘Ž) ∈ dom 𝑧 ↔ suc (rankβ€˜π‘Ž) ∈ dom 𝑧))
4639, 45mpbid 231 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ suc (rankβ€˜π‘Ž) ∈ dom 𝑧)
4730, 37, 46rspcdva 3583 . . 3 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ (π‘§β€˜suc (rankβ€˜π‘Ž)) We (𝑅1β€˜suc (rankβ€˜π‘Ž)))
48 wess 5621 . . 3 ({𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)} βŠ† (𝑅1β€˜suc (rankβ€˜π‘Ž)) β†’ ((π‘§β€˜suc (rankβ€˜π‘Ž)) We (𝑅1β€˜suc (rankβ€˜π‘Ž)) β†’ (π‘§β€˜suc (rankβ€˜π‘Ž)) We {𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)}))
4927, 47, 48sylc 65 . 2 ((πœ‘ ∧ π‘Ž ∈ (𝑅1β€˜dom 𝑧)) β†’ (π‘§β€˜suc (rankβ€˜π‘Ž)) We {𝑏 ∈ (𝑅1β€˜dom 𝑧) ∣ (rankβ€˜π‘) = (rankβ€˜π‘Ž)})
50 rankf 9731 . . . 4 rank:βˆͺ (𝑅1 β€œ On)⟢On
5150a1i 11 . . 3 (πœ‘ β†’ rank:βˆͺ (𝑅1 β€œ On)⟢On)
5251, 14fssresd 6710 . 2 (πœ‘ β†’ (rank β†Ύ (𝑅1β€˜dom 𝑧)):(𝑅1β€˜dom 𝑧)⟢On)
53 epweon 7710 . . 3 E We On
5453a1i 11 . 2 (πœ‘ β†’ E We On)
552, 3, 49, 52, 54fnwe2 41383 1 (πœ‘ β†’ 𝐹 We (𝑅1β€˜dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  βˆͺ cuni 4866   class class class wbr 5106  {copab 5168   E cep 5537   We wwe 5588  dom cdm 5634   β€œ cima 5637  Ord word 6317  Oncon0 6318  suc csuc 6320  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  π‘…1cr1 9699  rankcrnk 9700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-r1 9701  df-rank 9702
This theorem is referenced by:  aomclem5  41388
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