MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infpssrlem3 Structured version   Visualization version   GIF version

Theorem infpssrlem3 9165
Description: Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.)
Hypotheses
Ref Expression
infpssrlem.a (𝜑𝐵𝐴)
infpssrlem.c (𝜑𝐹:𝐵1-1-onto𝐴)
infpssrlem.d (𝜑𝐶 ∈ (𝐴𝐵))
infpssrlem.e 𝐺 = (rec(𝐹, 𝐶) ↾ ω)
Assertion
Ref Expression
infpssrlem3 (𝜑𝐺:ω⟶𝐴)

Proof of Theorem infpssrlem3
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 7575 . . . 4 (rec(𝐹, 𝐶) ↾ ω) Fn ω
2 infpssrlem.e . . . . 5 𝐺 = (rec(𝐹, 𝐶) ↾ ω)
32fneq1i 6023 . . . 4 (𝐺 Fn ω ↔ (rec(𝐹, 𝐶) ↾ ω) Fn ω)
41, 3mpbir 221 . . 3 𝐺 Fn ω
54a1i 11 . 2 (𝜑𝐺 Fn ω)
6 fveq2 6229 . . . . . 6 (𝑐 = ∅ → (𝐺𝑐) = (𝐺‘∅))
76eleq1d 2715 . . . . 5 (𝑐 = ∅ → ((𝐺𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8 fveq2 6229 . . . . . 6 (𝑐 = 𝑏 → (𝐺𝑐) = (𝐺𝑏))
98eleq1d 2715 . . . . 5 (𝑐 = 𝑏 → ((𝐺𝑐) ∈ 𝐴 ↔ (𝐺𝑏) ∈ 𝐴))
10 fveq2 6229 . . . . . 6 (𝑐 = suc 𝑏 → (𝐺𝑐) = (𝐺‘suc 𝑏))
1110eleq1d 2715 . . . . 5 (𝑐 = suc 𝑏 → ((𝐺𝑐) ∈ 𝐴 ↔ (𝐺‘suc 𝑏) ∈ 𝐴))
12 infpssrlem.a . . . . . . 7 (𝜑𝐵𝐴)
13 infpssrlem.c . . . . . . 7 (𝜑𝐹:𝐵1-1-onto𝐴)
14 infpssrlem.d . . . . . . 7 (𝜑𝐶 ∈ (𝐴𝐵))
1512, 13, 14, 2infpssrlem1 9163 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
1614eldifad 3619 . . . . . 6 (𝜑𝐶𝐴)
1715, 16eqeltrd 2730 . . . . 5 (𝜑 → (𝐺‘∅) ∈ 𝐴)
1812adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → 𝐵𝐴)
19 f1ocnv 6187 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
20 f1of 6175 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
2113, 19, 203syl 18 . . . . . . . . 9 (𝜑𝐹:𝐴𝐵)
2221ffvelrnda 6399 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐹‘(𝐺𝑏)) ∈ 𝐵)
2318, 22sseldd 3637 . . . . . . 7 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐹‘(𝐺𝑏)) ∈ 𝐴)
2412, 13, 14, 2infpssrlem2 9164 . . . . . . . 8 (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (𝐹‘(𝐺𝑏)))
2524eleq1d 2715 . . . . . . 7 (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (𝐹‘(𝐺𝑏)) ∈ 𝐴))
2623, 25syl5ibr 236 . . . . . 6 (𝑏 ∈ ω → ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴))
2726expd 451 . . . . 5 (𝑏 ∈ ω → (𝜑 → ((𝐺𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴)))
287, 9, 11, 17, 27finds2 7136 . . . 4 (𝑐 ∈ ω → (𝜑 → (𝐺𝑐) ∈ 𝐴))
2928com12 32 . . 3 (𝜑 → (𝑐 ∈ ω → (𝐺𝑐) ∈ 𝐴))
3029ralrimiv 2994 . 2 (𝜑 → ∀𝑐 ∈ ω (𝐺𝑐) ∈ 𝐴)
31 ffnfv 6428 . 2 (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺𝑐) ∈ 𝐴))
325, 30, 31sylanbrc 699 1 (𝜑𝐺:ω⟶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cdif 3604  wss 3607  c0 3948  ccnv 5142  cres 5145  suc csuc 5763   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  infpssrlem4  9166  infpssrlem5  9167
  Copyright terms: Public domain W3C validator