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Theorem rdgsucmpt2 8403
Description: This version of rdgsucmpt 8404 avoids the not-free hypothesis of rdgsucmptf 8401 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 = 𝑥𝐸 = 𝐶)
rdgsucmpt2.3 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
Assertion
Ref Expression
rdgsucmpt2 ((𝐵 ∈ On ∧ 𝐷𝑉) → (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
StepHypRef Expression
1 nfcv 2902 . 2 𝑦𝐴
2 nfcv 2902 . 2 𝑦𝐵
3 nfcv 2902 . 2 𝑦𝐷
4 rdgsucmpt2.1 . . 3 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5 rdgsucmpt2.2 . . . . 5 (𝑦 = 𝑥𝐸 = 𝐶)
65cbvmptv 5245 . . . 4 (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7 rdgeq1 8384 . . . 4 ((𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) → rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴))
86, 7ax-mp 5 . . 3 rec((𝑦 ∈ V ↦ 𝐸), 𝐴) = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
94, 8eqtr4i 2762 . 2 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10 rdgsucmpt2.3 . 2 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
111, 2, 3, 9, 10rdgsucmptf 8401 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3466  cmpt 5215  Oncon0 6344  suc csuc 6346  cfv 6523  reccrdg 8382
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5269  ax-sep 5283  ax-nul 5290  ax-pr 5411  ax-un 7699
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3372  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-pss 3954  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-tr 5250  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5615  df-we 5617  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6280  df-ord 6347  df-on 6348  df-lim 6349  df-suc 6350  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7387  df-2nd 7949  df-frecs 8239  df-wrecs 8270  df-recs 8344  df-rdg 8383
This theorem is referenced by: (None)
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