Theorem infpssrlem3 10383

Description: Lemma for infpssr 10386. (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 8443	. . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω
2		infpssrlem.e	. . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)
3	2	fneq1i 6636	. . . 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 6885	. . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
7	6	eleq1d 2846	. . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8		fveq2 6885	. . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
9	8	eleq1d 2846	. . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴))
10		fveq2 6885	. . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
11	10	eleq1d 2846	. . . . 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 10381	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
16	14	eldifad 3911	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
17	15, 16	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
18	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴)
19		f1ocnv 6837	. . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵)
20		f1of 6824	. . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵)
21	13, 19, 20	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵)
22	21	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵)
23	18, 22	sseldd 3932	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)
24	12, 13, 14, 2	infpssrlem2 10382	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏)))
25	24	eleq1d 2846	. . . . . . 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 421	. . . . 5 ⊢ (𝑏 ∈ ω → (𝜑 → ((𝐺‘𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴)))
28	7, 9, 11, 17, 27	finds2 7910	. . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴))
29	28	com12 33	. . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴))
30	29	ralrimiv 3154	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)
31		ffnfv 7119	. 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴))
32	5, 30, 31	sylanbrc 595	1 ⊢ (𝜑 → 𝐺:ω⟶𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ^◡ccnv 5650   ↾ cres 5653  suc csuc 6364   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  ωcom 7877  reccrdg 8417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418

This theorem is used by:  infpssrlem4  10384  infpssrlem5  10385