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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 8443 | . . 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 6893 | . . 3 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) |
4 | 2 | fveq1i 6893 | . . . 4 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵) |
5 | 4 | fveq2i 6895 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)) |
6 | 1, 3, 5 | 3eqtr4g 2790 | . 2 ⊢ (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵))) |
7 | fvex 6905 | . . 3 ⊢ (𝐹‘𝐵) ∈ V | |
8 | nfmpt1 5251 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶) | |
9 | rdgsucmptf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
10 | 8, 9 | nfrdg 8433 | . . . . . 6 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
11 | 2, 10 | nfcxfr 2890 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
12 | rdgsucmptf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
13 | 11, 12 | nffv 6902 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵) |
14 | rdgsucmptf.3 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
15 | rdgsucmptf.5 | . . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
16 | eqid 2725 | . . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶) | |
17 | 13, 14, 15, 16 | fvmptf 7021 | . . 3 ⊢ (((𝐹‘𝐵) ∈ V ∧ 𝐷 ∈ 𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
18 | 7, 17 | mpan 688 | . 2 ⊢ (𝐷 ∈ 𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = 𝐷) |
19 | 6, 18 | sylan9eq 2785 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Ⅎwnfc 2875 Vcvv 3463 ↦ cmpt 5226 Oncon0 6364 suc csuc 6366 ‘cfv 6543 reccrdg 8428 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
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 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 |
This theorem is referenced by: rdgsucmpt2 8449 rdgsucmpt 8450 ttrclselem1 9748 ttrclselem2 9749 rdgssun 36914 exrecfnlem 36915 |
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