Theorem rdgsucmptnf 8370

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class 𝐷 is a proper class). This is a technical lemma that can be used together with rdgsucmptf 8369 to help eliminate redundant sethood antecedents. (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
rdgsucmptnf	⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Proof of Theorem rdgsucmptnf

Step	Hyp	Ref	Expression
1		rdgsucmptf.4	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
2	1	fveq1i 6843	. 2 ⊢ (𝐹‘suc 𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵)
3		rdgdmlim 8358	. . . . 5 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
4		limsuc 7801	. . . . 5 ⊢ (Lim dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↔ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
6		rdgsucg 8364	. . . . . . 7 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)))
7	1	fveq1i 6843	. . . . . . . 8 ⊢ (𝐹‘𝐵) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵)
8	7	fveq2i 6845	. . . . . . 7 ⊢ ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ((𝑥 ∈ V ↦ 𝐶)‘(rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘𝐵))
9	6, 8	eqtr4di 2790	. . . . . 6 ⊢ (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)))
10		nfmpt1 5199	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ V ↦ 𝐶)
11		rdgsucmptf.1	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
12	10, 11	nfrdg 8355	. . . . . . . . 9 ⊢ Ⅎ𝑥rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
13	1, 12	nfcxfr 2897	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
14		rdgsucmptf.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
15	13, 14	nffv 6852	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝐵)
16		rdgsucmptf.3	. . . . . . 7 ⊢ Ⅎ𝑥𝐷
17		rdgsucmptf.5	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → 𝐶 = 𝐷)
18		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ V ↦ 𝐶) = (𝑥 ∈ V ↦ 𝐶)
19	15, 16, 17, 18	fvmptnf 6972	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → ((𝑥 ∈ V ↦ 𝐶)‘(𝐹‘𝐵)) = ∅)
20	9, 19	sylan9eqr 2794	. . . . 5 ⊢ ((¬ 𝐷 ∈ V ∧ 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴)) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
21	20	ex 412	. . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
22	5, 21	biimtrrid 243	. . 3 ⊢ (¬ 𝐷 ∈ V → (suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅))
23		ndmfv 6874	. . 3 ⊢ (¬ suc 𝐵 ∈ dom rec((𝑥 ∈ V ↦ 𝐶), 𝐴) → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
24	22, 23	pm2.61d1 180	. 2 ⊢ (¬ 𝐷 ∈ V → (rec((𝑥 ∈ V ↦ 𝐶), 𝐴)‘suc 𝐵) = ∅)
25	2, 24	eqtrid 2784	1 ⊢ (¬ 𝐷 ∈ V → (𝐹‘suc 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  Vcvv 3442  ∅c0 4287   ↦ cmpt 5181  dom cdm 5632  Lim wlim 6326  suc csuc 6327  ‘cfv 6500  reccrdg 8350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351

This theorem is referenced by:  ttrclselem1  9646