![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj60 | Structured version Visualization version GIF version |
Description: Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj60.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj60.2 | ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj60.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj60.4 | ⊢ 𝐹 = ∪ 𝐶 |
Ref | Expression |
---|---|
bnj60 | ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj60.1 | . . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
2 | bnj60.2 | . . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
3 | bnj60.3 | . . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
4 | 1, 2, 3 | bnj1497 34691 | . . . 4 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔 |
5 | eqid 2728 | . . . . . . . 8 ⊢ (dom 𝑔 ∩ dom ℎ) = (dom 𝑔 ∩ dom ℎ) | |
6 | 1, 2, 3, 5 | bnj1311 34655 | . . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶) → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) |
7 | 6 | 3expia 1119 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → (ℎ ∈ 𝐶 → (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))) |
8 | 7 | ralrimiv 3142 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶) → ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) |
9 | 8 | ralrimiva 3143 | . . . 4 ⊢ (𝑅 FrSe 𝐴 → ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) |
10 | biid 261 | . . . . 5 ⊢ (∀𝑔 ∈ 𝐶 Fun 𝑔 ↔ ∀𝑔 ∈ 𝐶 Fun 𝑔) | |
11 | biid 261 | . . . . 5 ⊢ ((∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) ↔ (∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ)))) | |
12 | 10, 5, 11 | bnj1383 34462 | . . . 4 ⊢ ((∀𝑔 ∈ 𝐶 Fun 𝑔 ∧ ∀𝑔 ∈ 𝐶 ∀ℎ ∈ 𝐶 (𝑔 ↾ (dom 𝑔 ∩ dom ℎ)) = (ℎ ↾ (dom 𝑔 ∩ dom ℎ))) → Fun ∪ 𝐶) |
13 | 4, 9, 12 | sylancr 586 | . . 3 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐶) |
14 | bnj60.4 | . . . 4 ⊢ 𝐹 = ∪ 𝐶 | |
15 | 14 | funeqi 6574 | . . 3 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐶) |
16 | 13, 15 | sylibr 233 | . 2 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹) |
17 | 1, 2, 3, 14 | bnj1498 34692 | . 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴) |
18 | 16, 17 | bnj1422 34468 | 1 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 ∀wral 3058 ∃wrex 3067 ∩ cin 3946 ⊆ wss 3947 ⟨cop 4635 ∪ cuni 4908 dom cdm 5678 ↾ cres 5680 Fun wfun 6542 Fn wfn 6543 ‘cfv 6548 predc-bnj14 34319 FrSe w-bnj15 34323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-reg 9615 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8486 df-bnj17 34318 df-bnj14 34320 df-bnj13 34322 df-bnj15 34324 df-bnj18 34326 df-bnj19 34328 |
This theorem is referenced by: bnj1501 34698 bnj1523 34702 |
Copyright terms: Public domain | W3C validator |