Theorem infpssrlem3 10306

Description: Lemma for infpssr 10309. (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 8441	. . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω
2		infpssrlem.e	. . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)
3	2	fneq1i 6646	. . . 4 ⊢ (𝐺 Fn ω ↔ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω)
4	1, 3	mpbir 230	. . 3 ⊢ 𝐺 Fn ω
5	4	a1i 11	. 2 ⊢ (𝜑 → 𝐺 Fn ω)
6		fveq2 6891	. . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
7	6	eleq1d 2817	. . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8		fveq2 6891	. . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
9	8	eleq1d 2817	. . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴))
10		fveq2 6891	. . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
11	10	eleq1d 2817	. . . . 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 10304	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
16	14	eldifad 3960	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
17	15, 16	eqeltrd 2832	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
18	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴)
19		f1ocnv 6845	. . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵)
20		f1of 6833	. . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵)
21	13, 19, 20	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵)
22	21	ffvelcdmda 7086	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵)
23	18, 22	sseldd 3983	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)
24	12, 13, 14, 2	infpssrlem2 10305	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏)))
25	24	eleq1d 2817	. . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴))
26	23, 25	imbitrrid 245	. . . . . 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 7895	. . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴))
29	28	com12 32	. . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴))
30	29	ralrimiv 3144	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)
31		ffnfv 7120	. 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴))
32	5, 30, 31	sylanbrc 582	1 ⊢ (𝜑 → 𝐺:ω⟶𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  ^◡ccnv 5675   ↾ cres 5678  suc csuc 6366   Fn wfn 6538  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  ωcom 7859  reccrdg 8415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416

This theorem is referenced by:  infpssrlem4  10307  infpssrlem5  10308