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Theorem rdgsucmptf 7693
 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.)
Hypotheses
Ref Expression
rdgsucmptf.1 𝑥𝐴
rdgsucmptf.2 𝑥𝐵
rdgsucmptf.3 𝑥𝐷
rdgsucmptf.4 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
rdgsucmptf.5 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
rdgsucmptf ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem rdgsucmptf
StepHypRef Expression
1 rdgsuc 7689 . . 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)
32fveq1i 6353 . . 3 (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
42fveq1i 6353 . . . 4 (𝐹𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
54fveq2i 6355 . . 3 ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
61, 3, 53eqtr4g 2819 . 2 (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)))
7 fvex 6362 . . 3 (𝐹𝐵) ∈ V
8 nfmpt1 4899 . . . . . . 7 𝑥(𝑥 ∈ V ↦ 𝐶)
9 rdgsucmptf.1 . . . . . . 7 𝑥𝐴
108, 9nfrdg 7679 . . . . . 6 𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
112, 10nfcxfr 2900 . . . . 5 𝑥𝐹
12 rdgsucmptf.2 . . . . 5 𝑥𝐵
1311, 12nffv 6359 . . . 4 𝑥(𝐹𝐵)
14 rdgsucmptf.3 . . . 4 𝑥𝐷
15 rdgsucmptf.5 . . . 4 (𝑥 = (𝐹𝐵) → 𝐶 = 𝐷)
16 eqid 2760 . . . 4 (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
1713, 14, 15, 16fvmptf 6463 . . 3 (((𝐹𝐵) ∈ V ∧ 𝐷𝑉) → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
187, 17mpan 708 . 2 (𝐷𝑉 → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹𝐵)) = 𝐷)
196, 18sylan9eq 2814 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Ⅎwnfc 2889  Vcvv 3340   ↦ cmpt 4881  Oncon0 5884  suc csuc 5886  ‘cfv 6049  reccrdg 7674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7576  df-recs 7637  df-rdg 7675 This theorem is referenced by:  rdgsucmpt2  7695  rdgsucmpt  7696
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