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Theorem infpssrlem3 9725
Description: Lemma for infpssr 9728. (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 8066 . . . 4 (rec(𝐹, 𝐶) ↾ ω) Fn ω
2 infpssrlem.e . . . . 5 𝐺 = (rec(𝐹, 𝐶) ↾ ω)
32fneq1i 6438 . . . 4 (𝐺 Fn ω ↔ (rec(𝐹, 𝐶) ↾ ω) Fn ω)
41, 3mpbir 234 . . 3 𝐺 Fn ω
54a1i 11 . 2 (𝜑𝐺 Fn ω)
6 fveq2 6661 . . . . . 6 (𝑐 = ∅ → (𝐺𝑐) = (𝐺‘∅))
76eleq1d 2900 . . . . 5 (𝑐 = ∅ → ((𝐺𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8 fveq2 6661 . . . . . 6 (𝑐 = 𝑏 → (𝐺𝑐) = (𝐺𝑏))
98eleq1d 2900 . . . . 5 (𝑐 = 𝑏 → ((𝐺𝑐) ∈ 𝐴 ↔ (𝐺𝑏) ∈ 𝐴))
10 fveq2 6661 . . . . . 6 (𝑐 = suc 𝑏 → (𝐺𝑐) = (𝐺‘suc 𝑏))
1110eleq1d 2900 . . . . 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 9723 . . . . . 6 (𝜑 → (𝐺‘∅) = 𝐶)
1614eldifad 3931 . . . . . 6 (𝜑𝐶𝐴)
1715, 16eqeltrd 2916 . . . . 5 (𝜑 → (𝐺‘∅) ∈ 𝐴)
1812adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → 𝐵𝐴)
19 f1ocnv 6618 . . . . . . . . . 10 (𝐹:𝐵1-1-onto𝐴𝐹:𝐴1-1-onto𝐵)
20 f1of 6606 . . . . . . . . . 10 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
2113, 19, 203syl 18 . . . . . . . . 9 (𝜑𝐹:𝐴𝐵)
2221ffvelrnda 6842 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐹‘(𝐺𝑏)) ∈ 𝐵)
2318, 22sseldd 3954 . . . . . . 7 ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐹‘(𝐺𝑏)) ∈ 𝐴)
2412, 13, 14, 2infpssrlem2 9724 . . . . . . . 8 (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (𝐹‘(𝐺𝑏)))
2524eleq1d 2900 . . . . . . 7 (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (𝐹‘(𝐺𝑏)) ∈ 𝐴))
2623, 25syl5ibr 249 . . . . . 6 (𝑏 ∈ ω → ((𝜑 ∧ (𝐺𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴))
2726expd 419 . . . . 5 (𝑏 ∈ ω → (𝜑 → ((𝐺𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴)))
287, 9, 11, 17, 27finds2 7605 . . . 4 (𝑐 ∈ ω → (𝜑 → (𝐺𝑐) ∈ 𝐴))
2928com12 32 . . 3 (𝜑 → (𝑐 ∈ ω → (𝐺𝑐) ∈ 𝐴))
3029ralrimiv 3176 . 2 (𝜑 → ∀𝑐 ∈ ω (𝐺𝑐) ∈ 𝐴)
31 ffnfv 6873 . 2 (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺𝑐) ∈ 𝐴))
325, 30, 31sylanbrc 586 1 (𝜑𝐺:ω⟶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  cdif 3916  wss 3919  c0 4276  ccnv 5541  cres 5544  suc csuc 6180   Fn wfn 6338  wf 6339  1-1-ontowf1o 6342  cfv 6343  ωcom 7574  reccrdg 8041
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042
This theorem is referenced by:  infpssrlem4  9726  infpssrlem5  9727
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