Theorem rdgsucmpt2 8422

Description: This version of rdgsucmpt 8423 avoids the not-free hypothesis of rdgsucmptf 8420 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 2923	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2923	. 2 ⊢ Ⅎ𝑦𝐵
3		nfcv 2923	. 2 ⊢ Ⅎ𝑦𝐷
4		rdgsucmpt2.1	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5		rdgsucmpt2.2	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
6	5	cbvmptv 5209	. . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7		rdgeq1 8403	. . . 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 2787	. 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10		rdgsucmpt2.3	. 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)
11	1, 2, 3, 9, 10	rdgsucmptf 8420	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ↦ cmpt 5186  Oncon0 6355  suc csuc 6357  ‘cfv 6531  reccrdg 8401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402

This theorem is used by:  ttctr  37251  ttcmin  37254  dfttc2g  37264