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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1384 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj60 32336. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1384.1 | ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} |
bnj1384.2 | ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 |
bnj1384.3 | ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} |
bnj1384.4 | ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) |
bnj1384.5 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} |
bnj1384.6 | ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) |
bnj1384.7 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) |
bnj1384.8 | ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) |
bnj1384.9 | ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} |
bnj1384.10 | ⊢ 𝑃 = ∪ 𝐻 |
Ref | Expression |
---|---|
bnj1384 | ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1384.1 | . . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)} | |
2 | bnj1384.2 | . . . . 5 ⊢ 𝑌 = 〈𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))〉 | |
3 | bnj1384.3 | . . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
4 | bnj1384.4 | . . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) | |
5 | bnj1384.5 | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} | |
6 | bnj1384.6 | . . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)) | |
7 | bnj1384.7 | . . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)) | |
8 | bnj1384.8 | . . . . 5 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏) | |
9 | bnj1384.9 | . . . . 5 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
10 | bnj1384.10 | . . . . 5 ⊢ 𝑃 = ∪ 𝐻 | |
11 | 1, 2, 3, 4, 8 | bnj1373 32304 | . . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | bnj1371 32303 | . . . 4 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓) |
13 | 12 | rgen 3150 | . . 3 ⊢ ∀𝑓 ∈ 𝐻 Fun 𝑓 |
14 | id 22 | . . . . . 6 ⊢ (𝑅 FrSe 𝐴 → 𝑅 FrSe 𝐴) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | bnj1374 32305 | . . . . . 6 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) |
16 | nfab1 2981 | . . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′} | |
17 | 9, 16 | nfcxfr 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝐻 |
18 | 17 | nfcri 2973 | . . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐻 |
19 | nfab1 2981 | . . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))} | |
20 | 3, 19 | nfcxfr 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑓𝐶 |
21 | 20 | nfcri 2973 | . . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶 |
22 | 18, 21 | nfim 1897 | . . . . . . 7 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶) |
23 | eleq1w 2897 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻)) | |
24 | eleq1w 2897 | . . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶)) | |
25 | 23, 24 | imbi12d 347 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) ↔ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶))) |
26 | 22, 25, 15 | chvarfv 2242 | . . . . . 6 ⊢ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶) |
27 | eqid 2823 | . . . . . . 7 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔) | |
28 | 1, 2, 3, 27 | bnj1326 32300 | . . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) |
29 | 14, 15, 26, 28 | syl3an 1156 | . . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) |
30 | 29 | 3expib 1118 | . . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))) |
31 | 30 | ralrimivv 3192 | . . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) |
32 | biid 263 | . . . 4 ⊢ (∀𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀𝑓 ∈ 𝐻 Fun 𝑓) | |
33 | biid 263 | . . . 4 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) ↔ (∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))) | |
34 | 9 | bnj1317 32095 | . . . 4 ⊢ (𝑧 ∈ 𝐻 → ∀𝑓 𝑧 ∈ 𝐻) |
35 | 32, 27, 33, 34 | bnj1386 32107 | . . 3 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun ∪ 𝐻) |
36 | 13, 31, 35 | sylancr 589 | . 2 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐻) |
37 | 10 | funeqi 6378 | . 2 ⊢ (Fun 𝑃 ↔ Fun ∪ 𝐻) |
38 | 36, 37 | sylibr 236 | 1 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 [wsbc 3774 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 〈cop 4575 ∪ cuni 4840 class class class wbr 5068 dom cdm 5557 ↾ cres 5559 Fun wfun 6351 Fn wfn 6352 ‘cfv 6357 predc-bnj14 31960 FrSe w-bnj15 31964 trClc-bnj18 31966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-bnj17 31959 df-bnj14 31961 df-bnj13 31963 df-bnj15 31965 df-bnj18 31967 df-bnj19 31969 |
This theorem is referenced by: bnj1312 32332 |
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