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Mirrors > Home > MPE Home > Th. List > nfwrecsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of nfwrecs 8320 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 9-Jun-2018.) |
Ref | Expression |
---|---|
nfwrecsOLD.1 | ⊢ Ⅎ𝑥𝑅 |
nfwrecsOLD.2 | ⊢ Ⅎ𝑥𝐴 |
nfwrecsOLD.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nfwrecsOLD | ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfwrecsOLD 8317 | . 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))} | |
2 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑥 𝑓 Fn 𝑦 | |
3 | nfcv 2892 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
4 | nfwrecsOLD.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfss 3964 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴 |
6 | nfwrecsOLD.1 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑅 | |
7 | nfcv 2892 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 4, 7 | nfpred 6305 | . . . . . . . . 9 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) |
9 | 8, 3 | nfss 3964 | . . . . . . . 8 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
10 | 3, 9 | nfralw 3299 | . . . . . . 7 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦 |
11 | 5, 10 | nfan 1894 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) |
12 | nfwrecsOLD.3 | . . . . . . . . 9 ⊢ Ⅎ𝑥𝐹 | |
13 | nfcv 2892 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑓 | |
14 | 13, 8 | nfres 5981 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)) |
15 | 12, 14 | nffv 6902 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
16 | 15 | nfeq2 2910 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
17 | 3, 16 | nfralw 3299 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))) |
18 | 2, 11, 17 | nf3an 1896 | . . . . 5 ⊢ Ⅎ𝑥(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
19 | 18 | nfex 2312 | . . . 4 ⊢ Ⅎ𝑥∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
20 | 19 | nfab 2898 | . . 3 ⊢ Ⅎ𝑥{𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))} |
21 | 20 | nfuni 4910 | . 2 ⊢ Ⅎ𝑥∪ {𝑓 ∣ ∃𝑦(𝑓 Fn 𝑦 ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑦 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑦) ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑧))))} |
22 | 1, 21 | nfcxfr 2890 | 1 ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 {cab 2702 Ⅎwnfc 2875 ∀wral 3051 ⊆ wss 3939 ∪ cuni 4903 ↾ cres 5674 Predcpred 6299 Fn wfn 6538 ‘cfv 6543 wrecscwrecs 8315 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-ov 7419 df-2nd 7992 df-frecs 8285 df-wrecs 8316 |
This theorem is referenced by: (None) |
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