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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1500 | Structured version Visualization version GIF version |
Description: Well-founded recursion, part 2 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 |
---|---|
bnj1500.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1500.2 | ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ |
bnj1500.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1500.4 | ⊢ 𝐹 = ∪ 𝐶 |
Ref | Expression |
---|---|
bnj1500 | ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1500.1 | . 2 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
2 | bnj1500.2 | . 2 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩ | |
3 | bnj1500.3 | . 2 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
4 | bnj1500.4 | . 2 ⊢ 𝐹 = ∪ 𝐶 | |
5 | biid 260 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴)) | |
6 | biid 260 | . 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓)) | |
7 | biid 260 | . 2 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓) ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | bnj1501 34731 | 1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {cab 2705 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 ⟨cop 4638 ∪ cuni 4912 dom cdm 5682 ↾ cres 5684 Fn wfn 6548 ‘cfv 6553 predc-bnj14 34352 FrSe w-bnj15 34356 |
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-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9623 ax-inf2 9672 |
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-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-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 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-om 7877 df-1o 8493 df-bnj17 34351 df-bnj14 34353 df-bnj13 34355 df-bnj15 34357 df-bnj18 34359 df-bnj19 34361 |
This theorem is referenced by: bnj1523 34735 |
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