Theorem rdgsucmpt2 8369

Description: This version of rdgsucmpt 8370 avoids the not-free hypothesis of rdgsucmptf 8367 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 2898	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2898	. 2 ⊢ Ⅎ𝑦𝐵
3		nfcv 2898	. 2 ⊢ Ⅎ𝑦𝐷
4		rdgsucmpt2.1	. . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴)
5		rdgsucmpt2.2	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶)
6	5	cbvmptv 5189	. . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶)
7		rdgeq1 8350	. . . 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 2762	. 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴)
10		rdgsucmpt2.3	. 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)
11	1, 2, 3, 9, 10	rdgsucmptf 8367	1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ↦ cmpt 5166  Oncon0 6323  suc csuc 6325  ‘cfv 6498  reccrdg 8348

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349

This theorem is referenced by:  ttctr  36675  ttcmin  36678  dfttc2g  36688