Theorem rdgsucmptf 8058

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	⊢ Ⅎ𝑥𝐴
rdgsucmptf.2	⊢ Ⅎ𝑥𝐵
rdgsucmptf.3	⊢ Ⅎ𝑥𝐷
rdgsucmptf.4	⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
rdgsucmptf.5	⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
rdgsucmptf	⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Proof of Theorem rdgsucmptf

Step	Hyp	Ref	Expression
1		rdgsuc 8054	. . 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 6665	. . 3 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
4	2	fveq1i 6665	. . . 4 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
5	4	fveq2i 6667	. . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
6	1, 3, 5	3eqtr4g 2881	. 2 ⊢ (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
7		fvex 6677	. . 3 ⊢ (𝐹‘𝐵) ∈ V
8		nfmpt1 5156	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
9		rdgsucmptf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
10	8, 9	nfrdg 8044	. . . . . 6 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11	2, 10	nfcxfr 2975	. . . . 5 ⊢ Ⅎ𝑥𝐹
12		rdgsucmptf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
13	11, 12	nffv 6674	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵)
14		rdgsucmptf.3	. . . 4 ⊢ Ⅎ𝑥𝐷
15		rdgsucmptf.5	. . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
16		eqid 2821	. . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
17	13, 14, 15, 16	fvmptf 6783	. . 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 2876	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961  Vcvv 3494   ↦ cmpt 5138  Oncon0 6185  suc csuc 6187  ‘cfv 6349  reccrdg 8039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-wrecs 7941  df-recs 8002  df-rdg 8040

This theorem is referenced by:  rdgsucmpt2  8060  rdgsucmpt  8061  rdgssun  34653  exrecfnlem  34654