Theorem rdgsucmpt2 8451

Description: This version of rdgsucmpt 8452 avoids the not-free hypothesis of rdgsucmptf 8449 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
rdgsucmpt2.1	⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
rdgsucmpt2.2	⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
rdgsucmpt2.3	⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)

Assertion

Ref	Expression
rdgsucmpt2	⊢ ((𝐵 ∈ On ∧ 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 rdgsucmpt2

Step	Hyp	Ref	Expression
1		nfcv 2891	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2891	. 2 ⊢ Ⅎ𝑦𝐵
3		nfcv 2891	. 2 ⊢ Ⅎ𝑦𝐷
4		rdgsucmpt2.1	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5		rdgsucmpt2.2	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
6	5	cbvmptv 5262	. . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7		rdgeq1 8432	. . . 4 ⊢ ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
8	6, 7	ax-mp 5	. . 3 ⊢ rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
9	4, 8	eqtr4i 2756	. 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10		rdgsucmpt2.3	. 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)
11	1, 2, 3, 9, 10	rdgsucmptf 8449	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ↦ cmpt 5232  Oncon0 6371  suc csuc 6373  ‘cfv 6549  reccrdg 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431

This theorem is referenced by: (None)