Theorem rdgsucmpt2 8453

Description: This version of rdgsucmpt 8454 avoids the not-free hypothesis of rdgsucmptf 8451 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 2897	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐵
3		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐷
4		rdgsucmpt2.1	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5		rdgsucmpt2.2	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
6	5	cbvmptv 5237	. . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7		rdgeq1 8434	. . . 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 2760	. 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10		rdgsucmpt2.3	. 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)
11	1, 2, 3, 9, 10	rdgsucmptf 8451	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3464   ↦ cmpt 5207  Oncon0 6365  suc csuc 6367  ‘cfv 6542  reccrdg 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433

This theorem is referenced by: (None)