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Theorem rdgsucmpt2 7390
Description: This version of rdgsucmpt 7391 avoids the not-free hypothesis of rdgsucmptf 7388 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:   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 2750 . 2 𝑦𝐴
2 nfcv 2750 . 2 𝑦𝐵
3 nfcv 2750 . 2 𝑦𝐷
4 rdgsucmpt2.1 . . 3 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5 rdgsucmpt2.2 . . . . 5 (𝑦 = 𝑥𝐸 = 𝐶)
65cbvmptv 4672 . . . 4 (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7 rdgeq1 7371 . . . 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 2634 . 2 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10 rdgsucmpt2.3 . 2 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
111, 2, 3, 9, 10rdgsucmptf 7388 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637  Oncon0 5625  suc csuc 5627  cfv 5789  reccrdg 7369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-wrecs 7271  df-recs 7332  df-rdg 7370
This theorem is referenced by: (None)
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