Theorem frsucmpt2 8070

Description: The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation), using double-substitution instead of a bound variable condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
frsucmpt2.1	⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
frsucmpt2.2	⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
frsucmpt2.3	⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)

Assertion

Ref	Expression
frsucmpt2	⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝐷   𝑥,𝐸

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝐸(𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem frsucmpt2

Step	Hyp	Ref	Expression
1		nfcv 2977	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2977	. 2 ⊢ Ⅎ𝑦𝐵
3		nfcv 2977	. 2 ⊢ Ⅎ𝑦𝐷
4		frsucmpt2.1	. . 3 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
5		frsucmpt2.2	. . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
6	5	cbvmptv 5162	. . . . 5 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7		rdgeq1 8041	. . . . 5 ⊢ ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
8	6, 7	ax-mp 5	. . . 4 ⊢ rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
9	8	reseq1i 5844	. . 3 ⊢ (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω) = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)
10	4, 9	eqtr4i 2847	. 2 ⊢ 𝐹 = (rec((𝑦 ∈ V ↦ 𝐸), 𝐴) ↾ ω)
11		frsucmpt2.3	. 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)
12	1, 2, 3, 10, 11	frsucmpt 8067	1 ⊢ ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3495   ↦ cmpt 5139   ↾ cres 5552  suc csuc 6188  ‘cfv 6350  ωcom 7574  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 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  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 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040

This theorem is referenced by:  unblem2  8765  unblem3  8766  inf0  9078  hsmexlem8  9840  wuncval2  10163  peano5nni  11635  peano2nn  11644  om2uzsuci  13310  neibastop2lem  33703