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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 7786 | . . 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 6434 | . . 3 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) |
4 | 2 | fveq1i 6434 | . . . 4 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵) |
5 | 4 | fveq2i 6436 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)) |
6 | 1, 3, 5 | 3eqtr4g 2886 | . 2 ⊢ (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
7 | fvex 6446 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
8 | nfmpt1 4970 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
9 | rdgsucmptf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
10 | 8, 9 | nfrdg 7776 | . . . . . 6 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
11 | 2, 10 | nfcxfr 2967 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
12 | rdgsucmptf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
13 | 11, 12 | nffv 6443 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
14 | rdgsucmptf.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
15 | rdgsucmptf.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
16 | eqid 2825 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
17 | 13, 14, 15, 16 | fvmptf 6548 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
18 | 7, 17 | mpan 683 | . 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
19 | 6, 18 | sylan9eq 2881 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Ⅎwnfc 2956 Vcvv 3414 ↦ cmpt 4952 Oncon0 5963 suc csuc 5965 ‘cfv 6123 reccrdg 7771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-wrecs 7672 df-recs 7734 df-rdg 7772 |
This theorem is referenced by: rdgsucmpt2 7792 rdgsucmpt 7793 |
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