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Mirrors > Home > MPE Home > Th. List > tfr3ALT | Structured version Visualization version GIF version |
Description: Alternate proof of tfr3 8346 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 7721 | . . . . . . 7 ⊢ (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥) | |
2 | 1 | reseq2d 5938 | . . . . . 6 ⊢ (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵 ↾ 𝑥)) |
3 | 2 | fveq2d 6847 | . . . . 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 3095 | . . 3 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) |
6 | epweon 7710 | . . . 4 ⊢ E We On | |
7 | epse 5617 | . . . 4 ⊢ E Se On | |
8 | tfrALT.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
9 | df-recs 8318 | . . . . . 6 ⊢ recs(𝐺) = wrecs( E , On, 𝐺) | |
10 | 8, 9 | eqtri 2765 | . . . . 5 ⊢ 𝐹 = wrecs( E , On, 𝐺) |
11 | 10 | wfr3 8284 | . . . 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 701 | . . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵) |
13 | 5, 12 | sylan2br 596 | . 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 E cep 5537 Se wse 5587 We wwe 5588 ↾ cres 5636 Predcpred 6253 Oncon0 6318 Fn wfn 6492 ‘cfv 6497 wrecscwrecs 8243 recscrecs 8317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 |
This theorem is referenced by: (None) |
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