Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infpssrlem3 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 9729. (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 8069 | . . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω | |
2 | infpssrlem.e | . . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) | |
3 | 2 | fneq1i 6449 | . . . 4 ⊢ (𝐺 Fn ω ↔ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 233 | . . 3 ⊢ 𝐺 Fn ω |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐺 Fn ω) |
6 | fveq2 6669 | . . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅)) | |
7 | 6 | eleq1d 2897 | . . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴)) |
8 | fveq2 6669 | . . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏)) | |
9 | 8 | eleq1d 2897 | . . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴)) |
10 | fveq2 6669 | . . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏)) | |
11 | 10 | eleq1d 2897 | . . . . 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 9724 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | 14 | eldifad 3947 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
17 | 15, 16 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴) |
18 | 12 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
19 | f1ocnv 6626 | . . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵) | |
20 | f1of 6614 | . . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵) | |
21 | 13, 19, 20 | 3syl 18 | . . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵) |
22 | 21 | ffvelrnda 6850 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵) |
23 | 18, 22 | sseldd 3967 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴) |
24 | 12, 13, 14, 2 | infpssrlem2 9725 | . . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏))) |
25 | 24 | eleq1d 2897 | . . . . . . 7 ⊢ (𝑏 ∈ ω → ((𝐺‘suc 𝑏) ∈ 𝐴 ↔ (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴)) |
26 | 23, 25 | syl5ibr 248 | . . . . . 6 ⊢ (𝑏 ∈ ω → ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (𝐺‘suc 𝑏) ∈ 𝐴)) |
27 | 26 | expd 418 | . . . . 5 ⊢ (𝑏 ∈ ω → (𝜑 → ((𝐺‘𝑏) ∈ 𝐴 → (𝐺‘suc 𝑏) ∈ 𝐴))) |
28 | 7, 9, 11, 17, 27 | finds2 7609 | . . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴)) |
29 | 28 | com12 32 | . . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴)) |
30 | 29 | ralrimiv 3181 | . 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴) |
31 | ffnfv 6881 | . 2 ⊢ (𝐺:ω⟶𝐴 ↔ (𝐺 Fn ω ∧ ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴)) | |
32 | 5, 30, 31 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐺:ω⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 ◡ccnv 5553 ↾ cres 5556 suc csuc 6192 Fn wfn 6349 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 ωcom 7579 reccrdg 8044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 |
This theorem is referenced by: infpssrlem4 9727 infpssrlem5 9728 |
Copyright terms: Public domain | W3C validator |