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Mirrors > Home > MPE Home > Th. List > tfr3ALT | Structured version Visualization version GIF version |
Description: Alternate proof of tfr3 8438 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 |
---|---|
tfr3ALT | ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predon 7805 | . . . . . . 7 ⊢ (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥) | |
2 | 1 | reseq2d 6000 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵 ↾ 𝑥)) |
3 | 2 | fveq2d 6911 | . . . . 5 ⊢ (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵 ↾ 𝑥))) |
4 | 3 | eqeq2d 2746 | . . . 4 ⊢ (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))) |
5 | 4 | ralbiia 3089 | . . 3 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) |
6 | epweon 7794 | . . . 4 ⊢ E We On | |
7 | epse 5671 | . . . 4 ⊢ E Se On | |
8 | tfrALT.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
9 | df-recs 8410 | . . . . . 6 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
10 | 8, 9 | eqtri 2763 | . . . . 5 ⊢ 𝐹 = wrecs( E , On, 𝐺) |
11 | 10 | wfr3 8376 | . . . 4 ⊢ ((( E We On ∧ E Se On) ∧ (𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))))) → 𝐹 = 𝐵) |
12 | 6, 7, 11 | mpanl12 702 | . . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵) |
13 | 5, 12 | sylan2br 595 | . 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐹 = 𝐵) |
14 | 13 | eqcomd 2741 | 1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 E cep 5588 Se wse 5639 We wwe 5640 ↾ cres 5691 Predcpred 6322 Oncon0 6386 Fn wfn 6558 ‘cfv 6563 wrecscwrecs 8335 recscrecs 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 |
This theorem is referenced by: (None) |
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