Theorem rdgsucmptf 8067

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 8063	. . 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 6674	. . 3 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
4	2	fveq1i 6674	. . . 4 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
5	4	fveq2i 6676	. . 3 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
6	1, 3, 5	3eqtr4g 2884	. 2 ⊢ (𝐵 ∈ On → (𝐹‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
7		fvex 6686	. . 3 ⊢ (𝐹‘𝐵) ∈ V
8		nfmpt1 5167	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
9		rdgsucmptf.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
10	8, 9	nfrdg 8053	. . . . . 6 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
11	2, 10	nfcxfr 2978	. . . . 5 ⊢ Ⅎ𝑥𝐹
12		rdgsucmptf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
13	11, 12	nffv 6683	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝐵)
14		rdgsucmptf.3	. . . 4 ⊢ Ⅎ𝑥𝐷
15		rdgsucmptf.5	. . . 4 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
16		eqid 2824	. . . 4 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
17	13, 14, 15, 16	fvmptf 6792	. . 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 2879	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Ⅎwnfc 2964  Vcvv 3497   ↦ cmpt 5149  Oncon0 6194  suc csuc 6196  ‘cfv 6358  reccrdg 8048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-wrecs 7950  df-recs 8011  df-rdg 8049

This theorem is referenced by:  rdgsucmpt2  8069  rdgsucmpt  8070  rdgssun  34663  exrecfnlem  34664