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Mirrors > Home > MPE Home > Th. List > rdgsucmptf | Structured version Visualization version GIF version |
Description: The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (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 |
---|---|
rdgsucmptf | ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsuc 8043 | . . 3 ⊢ (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))) | |
2 | rdgsucmptf.4 | . . . 4 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
3 | 2 | fveq1i 6646 | . . 3 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) |
4 | 2 | fveq1i 6646 | . . . 4 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵) |
5 | 4 | fveq2i 6648 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)) |
6 | 1, 3, 5 | 3eqtr4g 2858 | . 2 ⊢ (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
7 | fvex 6658 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
8 | nfmpt1 5128 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
9 | rdgsucmptf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
10 | 8, 9 | nfrdg 8033 | . . . . . 6 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
11 | 2, 10 | nfcxfr 2953 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
12 | rdgsucmptf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
13 | 11, 12 | nffv 6655 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
14 | rdgsucmptf.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
15 | rdgsucmptf.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
16 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
17 | 13, 14, 15, 16 | fvmptf 6766 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
18 | 7, 17 | mpan 689 | . 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
19 | 6, 18 | sylan9eq 2853 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Ⅎwnfc 2936 Vcvv 3441 ↦ cmpt 5110 Oncon0 6159 suc csuc 6161 ‘cfv 6324 reccrdg 8028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-wrecs 7930 df-recs 7991 df-rdg 8029 |
This theorem is referenced by: rdgsucmpt2 8049 rdgsucmpt 8050 rdgssun 34795 exrecfnlem 34796 |
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