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Mirrors > Home > MPE Home > Th. List > tfr2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of tfr2 8425 using well-ordered recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
tfrALT.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr2ALT | ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epweon 7783 | . . 3 ⊢ E We On | |
2 | epse 5665 | . . 3 ⊢ E Se On | |
3 | tfrALT.1 | . . . . 5 ⊢ 𝐹 = recs(𝐺) | |
4 | df-recs 8398 | . . . . 5 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
5 | 3, 4 | eqtri 2756 | . . . 4 ⊢ 𝐹 = wrecs( E , On, 𝐺) |
6 | 5 | wfr2 8363 | . . 3 ⊢ ((( E We On ∧ E Se On) ∧ 𝐴 ∈ On) → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴)))) |
7 | 1, 2, 6 | mpanl12 700 | . 2 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴)))) |
8 | predon 7794 | . . . 4 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴) | |
9 | 8 | reseq2d 5989 | . . 3 ⊢ (𝐴 ∈ On → (𝐹 ↾ Pred( E , On, 𝐴)) = (𝐹 ↾ 𝐴)) |
10 | 9 | fveq2d 6906 | . 2 ⊢ (𝐴 ∈ On → (𝐺‘(𝐹 ↾ Pred( E , On, 𝐴))) = (𝐺‘(𝐹 ↾ 𝐴))) |
11 | 7, 10 | eqtrd 2768 | 1 ⊢ (𝐴 ∈ On → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 E cep 5585 Se wse 5635 We wwe 5636 ↾ cres 5684 Predcpred 6309 Oncon0 6374 ‘cfv 6553 wrecscwrecs 8323 recscrecs 8397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 |
This theorem is referenced by: (None) |
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