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Mirrors > Home > MPE Home > Th. List > infpssrlem3 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 9168. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
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) |
Ref | Expression |
---|---|
infpssrlem3 | ⊢ (𝜑 → 𝐺:ω⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 7575 | . . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω | |
2 | infpssrlem.e | . . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) | |
3 | 2 | fneq1i 6023 | . . . 4 ⊢ (𝐺 Fn ω ↔ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 221 | . . 3 ⊢ 𝐺 Fn ω |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐺 Fn ω) |
6 | fveq2 6229 | . . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅)) | |
7 | 6 | eleq1d 2715 | . . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴)) |
8 | fveq2 6229 | . . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏)) | |
9 | 8 | eleq1d 2715 | . . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴)) |
10 | fveq2 6229 | . . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏)) | |
11 | 10 | eleq1d 2715 | . . . . 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 9163 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | 14 | eldifad 3619 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
17 | 15, 16 | eqeltrd 2730 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴) |
18 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
19 | f1ocnv 6187 | . . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵) | |
20 | f1of 6175 | . . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵) | |
21 | 13, 19, 20 | 3syl 18 | . . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵) |
22 | 21 | ffvelrnda 6399 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵) |
23 | 18, 22 | sseldd 3637 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴) |
24 | 12, 13, 14, 2 | infpssrlem2 9164 | . . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏))) |
25 | 24 | eleq1d 2715 | . . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)) |
26 | 23, 25 | syl5ibr 236 | . . . . . 6 ⊢ (𝑏 ∈ ω → ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴)) |
27 | 26 | expd 451 | . . . . 5 ⊢ (𝑏 ∈ ω → (𝜑 → ((𝐺‘𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴))) |
28 | 7, 9, 11, 17, 27 | finds2 7136 | . . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴)) |
29 | 28 | com12 32 | . . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴)) |
30 | 29 | ralrimiv 2994 | . 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴) |
31 | ffnfv 6428 | . 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)) | |
32 | 5, 30, 31 | sylanbrc 699 | 1 ⊢ (𝜑 → 𝐺:ω⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 ∖ cdif 3604 ⊆ wss 3607 ∅c0 3948 ◡ccnv 5142 ↾ cres 5145 suc csuc 5763 Fn wfn 5921 ⟶wf 5922 –1-1-onto→wf1o 5925 ‘cfv 5926 ωcom 7107 reccrdg 7550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 |
This theorem is referenced by: infpssrlem4 9166 infpssrlem5 9167 |
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