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Mirrors > Home > MPE Home > Th. List > rdgsucmptnf | Structured version Visualization version GIF version |
Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with rdgsucmptf 8449 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
rdgsucmptf.1 | ⊢ Ⅎ𝑥𝐴 |
rdgsucmptf.2 | ⊢ Ⅎ𝑥𝐵 |
rdgsucmptf.3 | ⊢ Ⅎ𝑥𝐷 |
rdgsucmptf.4 | ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
rdgsucmptf.5 | ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
rdgsucmptnf | ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsucmptf.4 | . . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
2 | 1 | fveq1i 6897 | . 2 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) |
3 | rdgdmlim 8438 | . . . . 5 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
4 | limsuc 7854 | . . . . 5 ⊢ (Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) |
6 | rdgsucg 8444 | . . . . . . 7 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))) | |
7 | 1 | fveq1i 6897 | . . . . . . . 8 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵) |
8 | 7 | fveq2i 6899 | . . . . . . 7 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)) |
9 | 6, 8 | eqtr4di 2783 | . . . . . 6 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
10 | nfmpt1 5257 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
11 | rdgsucmptf.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴 | |
12 | 10, 11 | nfrdg 8435 | . . . . . . . . 9 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
13 | 1, 12 | nfcxfr 2889 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 |
14 | rdgsucmptf.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
15 | 13, 14 | nffv 6906 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
16 | rdgsucmptf.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝐷 | |
17 | rdgsucmptf.5 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
18 | eqid 2725 | . . . . . . 7 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
19 | 15, 16, 17, 18 | fvmptnf 7026 | . . . . . 6 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅) |
20 | 9, 19 | sylan9eqr 2787 | . . . . 5 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) |
21 | 20 | ex 411 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)) |
22 | 5, 21 | biimtrrid 242 | . . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)) |
23 | ndmfv 6931 | . . 3 ⊢ (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) | |
24 | 22, 23 | pm2.61d1 180 | . 2 ⊢ (¬ 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) |
25 | 2, 24 | eqtrid 2777 | 1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 Vcvv 3461 ∅c0 4322 ↦ cmpt 5232 dom cdm 5678 Lim wlim 6372 suc csuc 6373 ‘cfv 6549 reccrdg 8430 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 |
This theorem is referenced by: ttrclselem1 9755 |
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