Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rdgsucmpt | 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 Mario Carneiro, 9-Sep-2013.) |
Ref | Expression |
---|---|
rdgsucmpt.1 | ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
rdgsucmpt.2 | ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
rdgsucmpt | ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2979 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2979 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfcv 2979 | . 2 ⊢ Ⅎ𝑥𝐷 | |
4 | rdgsucmpt.1 | . 2 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
5 | rdgsucmpt.2 | . 2 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷) | |
6 | 1, 2, 3, 4, 5 | rdgsucmptf 8066 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 Oncon0 6193 suc csuc 6195 ‘cfv 6357 reccrdg 8047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |