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Theorem rdgsucmpt2 8377
Description: This version of rdgsucmpt 8378 avoids the not-free hypothesis of rdgsucmptf 8375 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 2908 . 2 𝑦𝐴
2 nfcv 2908 . 2 𝑦𝐵
3 nfcv 2908 . 2 𝑦𝐷
4 rdgsucmpt2.1 . . 3 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5 rdgsucmpt2.2 . . . . 5 (𝑦 = 𝑥𝐸 = 𝐶)
65cbvmptv 5219 . . . 4 (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7 rdgeq1 8358 . . . 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 2768 . 2 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10 rdgsucmpt2.3 . 2 (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)
111, 2, 3, 9, 10rdgsucmptf 8375 1 ((𝐵 ∈ On ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3446  cmpt 5189  Oncon0 6318  suc csuc 6320  cfv 6497  reccrdg 8356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357
This theorem is referenced by: (None)
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