Theorem infpssrlem3 10289

Description: Lemma for infpssr 10292. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Hypotheses

Ref	Expression
infpssrlem.a	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
infpssrlem.c	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)
infpssrlem.d	⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
infpssrlem.e	⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)

Assertion

Ref	Expression
infpssrlem3	⊢ (𝜑 → 𝐺:ω⟶𝐴)

Proof of Theorem infpssrlem3

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frfnom 8422	. . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω
2		infpssrlem.e	. . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)
3	2	fneq1i 6633	. . . 4 ⊢ (𝐺 Fn ω ↔ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω)
4	1, 3	mpbir 234	. . 3 ⊢ 𝐺 Fn ω
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐺 Fn ω)
6		fveq2 6882	. . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
7	6	eleq1d 2854	. . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8		fveq2 6882	. . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
9	8	eleq1d 2854	. . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴))
10		fveq2 6882	. . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
11	10	eleq1d 2854	. . . . 5 ⊢ (𝑐 = suc 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘suc 𝑏) ∈ 𝐴))
12		infpssrlem.a	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
13		infpssrlem.c	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐴)
14		infpssrlem.d	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ∖ 𝐵))
15	12, 13, 14, 2	infpssrlem1 10287	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
16	14	eldifad 3925	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
17	15, 16	eqeltrd 2869	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
18	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴)
19		f1ocnv 6834	. . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵)
20		f1of 6821	. . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵)
21	13, 19, 20	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵)
22	21	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵)
23	18, 22	sseldd 3946	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)
24	12, 13, 14, 2	infpssrlem2 10288	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏)))
25	24	eleq1d 2854	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴))
26	23, 25	imbitrrid 249	. . . . . 6 ⊢ (𝑏 ∈ ω → ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴))
27	26	expd 420	. . . . 5 ⊢ (𝑏 ∈ ω → (𝜑 → ((𝐺‘𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴)))
28	7, 9, 11, 17, 27	finds2 7895	. . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴))
29	28	com12 33	. . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴))
30	29	ralrimiv 3162	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)
31		ffnfv 7115	. 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴))
32	5, 30, 31	sylanbrc 594	1 ⊢ (𝜑 → 𝐺:ω⟶𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ^◡ccnv 5661   ↾ cres 5664  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  ωcom 7862  reccrdg 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397

This theorem is referenced by:  infpssrlem4  10290  infpssrlem5  10291