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Mirrors > Home > MPE Home > Th. List > infpssrlem3 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 10309. (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 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) |
Colors of variables: wff setvar class |
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 |
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