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| Mirrors > Home > MPE Home > Th. List > tfr3ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of tfr3 8439 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 7806 | . . . . . . 7 ⊢ (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥) | |
| 2 | 1 | reseq2d 5997 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵 ↾ 𝑥)) |
| 3 | 2 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵 ↾ 𝑥))) |
| 4 | 3 | eqeq2d 2748 | . . . 4 ⊢ (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))) |
| 5 | 4 | ralbiia 3091 | . . 3 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) |
| 6 | epweon 7795 | . . . 4 ⊢ E We On | |
| 7 | epse 5667 | . . . 4 ⊢ E Se On | |
| 8 | tfrALT.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
| 9 | df-recs 8411 | . . . . . 6 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
| 10 | 8, 9 | eqtri 2765 | . . . . 5 ⊢ 𝐹 = wrecs( E , On, 𝐺) |
| 11 | 10 | wfr3 8377 | . . . 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 2743 | 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 1540 ∈ wcel 2108 ∀wral 3061 E cep 5583 Se wse 5635 We wwe 5636 ↾ cres 5687 Predcpred 6320 Oncon0 6384 Fn wfn 6556 ‘cfv 6561 wrecscwrecs 8336 recscrecs 8410 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 |
| This theorem is referenced by: (None) |
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