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Theorem indcardi 9456
Description: Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Hypotheses
Ref Expression
indcardi.a (𝜑𝐴𝑉)
indcardi.b (𝜑𝑇 ∈ dom card)
indcardi.c ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
indcardi.d (𝑥 = 𝑦 → (𝜓𝜒))
indcardi.e (𝑥 = 𝐴 → (𝜓𝜃))
indcardi.f (𝑥 = 𝑦𝑅 = 𝑆)
indcardi.g (𝑥 = 𝐴𝑅 = 𝑇)
Assertion
Ref Expression
indcardi (𝜑𝜃)
Distinct variable groups:   𝑥,𝑦,𝑇   𝑥,𝐴   𝑥,𝑆   𝜒,𝑥   𝜑,𝑥,𝑦   𝜃,𝑥   𝑦,𝑅   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem indcardi
StepHypRef Expression
1 indcardi.b . . 3 (𝜑𝑇 ∈ dom card)
2 domrefg 8531 . . 3 (𝑇 ∈ dom card → 𝑇𝑇)
31, 2syl 17 . 2 (𝜑𝑇𝑇)
4 indcardi.a . . 3 (𝜑𝐴𝑉)
5 cardon 9361 . . . 4 (card‘𝑇) ∈ On
65a1i 11 . . 3 (𝜑 → (card‘𝑇) ∈ On)
7 simpl1 1188 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜑)
8 simpr 488 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝑅𝑇)
9 simpr 488 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑅)
10 simpl1 1188 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝜑)
1110, 1syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑇 ∈ dom card)
12 sdomdom 8524 . . . . . . . . . . . . . . . 16 (𝑆𝑅𝑆𝑅)
13 simpl3 1190 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅𝑇)
14 domtr 8549 . . . . . . . . . . . . . . . 16 ((𝑆𝑅𝑅𝑇) → 𝑆𝑇)
1512, 13, 14syl2an2 685 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑇)
16 numdom 9453 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑆𝑇) → 𝑆 ∈ dom card)
1711, 15, 16syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆 ∈ dom card)
18 numdom 9453 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑅𝑇) → 𝑅 ∈ dom card)
1911, 13, 18syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅 ∈ dom card)
20 cardsdom2 9405 . . . . . . . . . . . . . 14 ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
2117, 19, 20syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
229, 21mpbird 260 . . . . . . . . . . . 12 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (card‘𝑆) ∈ (card‘𝑅))
23 id 22 . . . . . . . . . . . . 13 (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)))
2423com3l 89 . . . . . . . . . . . 12 ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2522, 15, 24sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒))
2625ex 416 . . . . . . . . . 10 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (𝑆𝑅 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2726com23 86 . . . . . . . . 9 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑆𝑅𝜒)))
2827alimdv 1917 . . . . . . . 8 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))
29283exp 1116 . . . . . . 7 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (𝑅𝑇 → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))))
3029com34 91 . . . . . 6 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑅𝑇 → ∀𝑦(𝑆𝑅𝜒)))))
31303imp1 1344 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → ∀𝑦(𝑆𝑅𝜒))
32 indcardi.c . . . . 5 ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
337, 8, 31, 32syl3anc 1368 . . . 4 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜓)
3433ex 416 . . 3 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) → (𝑅𝑇𝜓))
35 indcardi.f . . . . 5 (𝑥 = 𝑦𝑅 = 𝑆)
3635breq1d 5043 . . . 4 (𝑥 = 𝑦 → (𝑅𝑇𝑆𝑇))
37 indcardi.d . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
3836, 37imbi12d 348 . . 3 (𝑥 = 𝑦 → ((𝑅𝑇𝜓) ↔ (𝑆𝑇𝜒)))
39 indcardi.g . . . . 5 (𝑥 = 𝐴𝑅 = 𝑇)
4039breq1d 5043 . . . 4 (𝑥 = 𝐴 → (𝑅𝑇𝑇𝑇))
41 indcardi.e . . . 4 (𝑥 = 𝐴 → (𝜓𝜃))
4240, 41imbi12d 348 . . 3 (𝑥 = 𝐴 → ((𝑅𝑇𝜓) ↔ (𝑇𝑇𝜃)))
4335fveq2d 6653 . . 3 (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
4439fveq2d 6653 . . 3 (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
454, 6, 34, 38, 42, 43, 44tfisi 7557 . 2 (𝜑 → (𝑇𝑇𝜃))
463, 45mpd 15 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2112  wss 3884   class class class wbr 5033  dom cdm 5523  Oncon0 6163  cfv 6328  cdom 8494  csdm 8495  cardccrd 9352
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-wrecs 7934  df-recs 7995  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356
This theorem is referenced by:  uzindi  13349  symggen  18594
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