Theorem infpssrlem3 10227

Description: Lemma for infpssr 10230. (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 8376	. . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω
2		infpssrlem.e	. . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω)
3	2	fneq1i 6597	. . . 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 6842	. . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅))
7	6	eleq1d 2822	. . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴))
8		fveq2 6842	. . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏))
9	8	eleq1d 2822	. . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴))
10		fveq2 6842	. . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏))
11	10	eleq1d 2822	. . . . 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 10225	. . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
16	14	eldifad 3915	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
17	15, 16	eqeltrd 2837	. . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴)
18	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴)
19		f1ocnv 6794	. . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵)
20		f1of 6782	. . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵)
21	13, 19, 20	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵)
22	21	ffvelcdmda 7038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵)
23	18, 22	sseldd 3936	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)
24	12, 13, 14, 2	infpssrlem2 10226	. . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏)))
25	24	eleq1d 2822	. . . . . . 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 7850	. . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴))
29	28	com12 32	. . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴))
30	29	ralrimiv 3129	. 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)
31		ffnfv 7073	. 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴))
32	5, 30, 31	sylanbrc 584	1 ⊢ (𝜑 → 𝐺:ω⟶𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ^◡ccnv 5631   ↾ cres 5634  suc csuc 6327   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  ωcom 7818  reccrdg 8350

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351

This theorem is referenced by:  infpssrlem4  10228  infpssrlem5  10229