Theorem infpssrlem3 10258

Description: Lemma for infpssr 10261. (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 8403	. . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω
2		infpssrlem.e	. . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)
3	2	fneq1i 6615	. . . 4 ⊢ (𝐺 Fn ω ↔ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω)
4	1, 3	mpbir 231	. . 3 ⊢ 𝐺 Fn ω
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐺 Fn ω)
6		fveq2 6858	. . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
7	6	eleq1d 2813	. . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8		fveq2 6858	. . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
9	8	eleq1d 2813	. . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴))
10		fveq2 6858	. . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
11	10	eleq1d 2813	. . . . 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 10256	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
16	14	eldifad 3926	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
17	15, 16	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
18	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴)
19		f1ocnv 6812	. . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵)
20		f1of 6800	. . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵)
21	13, 19, 20	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵)
22	21	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵)
23	18, 22	sseldd 3947	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)
24	12, 13, 14, 2	infpssrlem2 10257	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏)))
25	24	eleq1d 2813	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴))
26	23, 25	imbitrrid 246	. . . . . 6 ⊢ (𝑏 ∈ ω → ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴))
27	26	expd 415	. . . . 5 ⊢ (𝑏 ∈ ω → (𝜑 → ((𝐺‘𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴)))
28	7, 9, 11, 17, 27	finds2 7874	. . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴))
29	28	com12 32	. . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴))
30	29	ralrimiv 3124	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)
31		ffnfv 7091	. 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴))
32	5, 30, 31	sylanbrc 583	1 ⊢ (𝜑 → 𝐺:ω⟶𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3911   ⊆ wss 3914  ∅c0 4296  ^◡ccnv 5637   ↾ cres 5640  suc csuc 6334   Fn wfn 6506  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  ωcom 7842  reccrdg 8377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378

This theorem is referenced by:  infpssrlem4  10259  infpssrlem5  10260