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Mirrors > Home > MPE Home > Th. List > rdgsucmpt2 | Structured version Visualization version GIF version |
Description: This version of rdgsucmpt 7527 avoids the not-free hypothesis of rdgsucmptf 7524 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
rdgsucmpt2.1 | ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) |
rdgsucmpt2.2 | ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) |
rdgsucmpt2.3 | ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) |
Ref | Expression |
---|---|
rdgsucmpt2 | ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐷 | |
4 | rdgsucmpt2.1 | . . 3 ⊢ 𝐹 = rec((𝑥 ∈ V ↦ 𝐶), 𝐴) | |
5 | rdgsucmpt2.2 | . . . . 5 ⊢ (𝑦 = 𝑥 → 𝐸 = 𝐶) | |
6 | 5 | cbvmptv 4750 | . . . 4 ⊢ (𝑦 ∈ V ↦ 𝐸) = (𝑥 ∈ V ↦ 𝐶) |
7 | rdgeq1 7507 | . . . 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 2647 | . 2 ⊢ 𝐹 = rec((𝑦 ∈ V ↦ 𝐸), 𝐴) |
10 | rdgsucmpt2.3 | . 2 ⊢ (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷) | |
11 | 1, 2, 3, 9, 10 | rdgsucmptf 7524 | 1 ⊢ ((𝐵 ∈ On ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 Oncon0 5723 suc csuc 5725 ‘cfv 5888 reccrdg 7505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 |
This theorem is referenced by: (None) |
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