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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 8153 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 6707 | . 2 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) |
3 | rdgdmlim 8142 | . . . . 5 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
4 | limsuc 7617 | . . . . 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 8148 | . . . . . . 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 6707 | . . . . . . . 8 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵) |
8 | 7 | fveq2i 6709 | . . . . . . 7 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)) |
9 | 6, 8 | eqtr4di 2792 | . . . . . 6 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
10 | nfmpt1 5142 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
11 | rdgsucmptf.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴 | |
12 | 10, 11 | nfrdg 8139 | . . . . . . . . 9 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
13 | 1, 12 | nfcxfr 2898 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 |
14 | rdgsucmptf.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐵 | |
15 | 13, 14 | nffv 6716 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
16 | rdgsucmptf.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝐷 | |
17 | rdgsucmptf.5 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
18 | eqid 2734 | . . . . . . 7 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
19 | 15, 16, 17, 18 | fvmptnf 6829 | . . . . . 6 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅) |
20 | 9, 19 | sylan9eqr 2796 | . . . . 5 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) |
21 | 20 | ex 416 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)) |
22 | 5, 21 | syl5bir 246 | . . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)) |
23 | ndmfv 6736 | . . 3 ⊢ (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) | |
24 | 22, 23 | pm2.61d1 183 | . 2 ⊢ (¬ 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅) |
25 | 2, 24 | syl5eq 2786 | 1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2880 Vcvv 3401 ∅c0 4227 ↦ cmpt 5124 dom cdm 5540 Lim wlim 6203 suc csuc 6204 ‘cfv 6369 reccrdg 8134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-wrecs 8036 df-recs 8097 df-rdg 8135 |
This theorem is referenced by: (None) |
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