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Mirrors > Home > MPE Home > Th. List > infpssrlem3 | Structured version Visualization version GIF version |
Description: Lemma for infpssr 10305. (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 8436 | . . . 4 ⊢ (rec(◡𝐹, 𝐶) ↾ ω) Fn ω | |
2 | infpssrlem.e | . . . . 5 ⊢ 𝐺 = (rec(◡𝐹, 𝐶) ↾ ω) | |
3 | 2 | fneq1i 6640 | . . . 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 6885 | . . . . . 6 ⊢ (𝑐 = ∅ → (𝐺‘𝑐) = (𝐺‘∅)) | |
7 | 6 | eleq1d 2812 | . . . . 5 ⊢ (𝑐 = ∅ → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘∅) ∈ 𝐴)) |
8 | fveq2 6885 | . . . . . 6 ⊢ (𝑐 = 𝑏 → (𝐺‘𝑐) = (𝐺‘𝑏)) | |
9 | 8 | eleq1d 2812 | . . . . 5 ⊢ (𝑐 = 𝑏 → ((𝐺‘𝑐) ∈ 𝐴 ↔ (𝐺‘𝑏) ∈ 𝐴)) |
10 | fveq2 6885 | . . . . . 6 ⊢ (𝑐 = suc 𝑏 → (𝐺‘𝑐) = (𝐺‘suc 𝑏)) | |
11 | 10 | eleq1d 2812 | . . . . 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 10300 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | 14 | eldifad 3955 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
17 | 15, 16 | eqeltrd 2827 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ 𝐴) |
18 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
19 | f1ocnv 6839 | . . . . . . . . . 10 ⊢ (𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐵) | |
20 | f1of 6827 | . . . . . . . . . 10 ⊢ (◡𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐴⟶𝐵) | |
21 | 13, 19, 20 | 3syl 18 | . . . . . . . . 9 ⊢ (𝜑 → ◡𝐹:𝐴⟶𝐵) |
22 | 21 | ffvelcdmda 7080 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐵) |
23 | 18, 22 | sseldd 3978 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐺‘𝑏) ∈ 𝐴) → (◡𝐹‘(𝐺‘𝑏)) ∈ 𝐴) |
24 | 12, 13, 14, 2 | infpssrlem2 10301 | . . . . . . . 8 ⊢ (𝑏 ∈ ω → (𝐺‘suc 𝑏) = (◡𝐹‘(𝐺‘𝑏))) |
25 | 24 | eleq1d 2812 | . . . . . . 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 7888 | . . . 4 ⊢ (𝑐 ∈ ω → (𝜑 → (𝐺‘𝑐) ∈ 𝐴)) |
29 | 28 | com12 32 | . . 3 ⊢ (𝜑 → (𝑐 ∈ ω → (𝐺‘𝑐) ∈ 𝐴)) |
30 | 29 | ralrimiv 3139 | . 2 ⊢ (𝜑 → ∀𝑐 ∈ ω (𝐺‘𝑐) ∈ 𝐴) |
31 | ffnfv 7114 | . 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 1533 ∈ wcel 2098 ∀wral 3055 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 ◡ccnv 5668 ↾ cres 5671 suc csuc 6360 Fn wfn 6532 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 ωcom 7852 reccrdg 8410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 |
This theorem is referenced by: infpssrlem4 10303 infpssrlem5 10304 |
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