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Theorem bnj1498 32941
Description: Technical lemma for bnj60 32942. 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.)
Hypotheses
Ref Expression
bnj1498.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1498.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1498.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1498.4 𝐹 = 𝐶
Assertion
Ref Expression
bnj1498 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓,𝑑)

Proof of Theorem bnj1498
Dummy variables 𝑡 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 4925 . . . . . . 7 (𝑧 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑧 ∈ dom 𝑓)
2 bnj1498.3 . . . . . . . . . . . . . . . 16 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1436 32719 . . . . . . . . . . . . . . 15 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
43bnj1299 32698 . . . . . . . . . . . . . 14 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
5 fndm 6520 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
64, 5bnj31 32598 . . . . . . . . . . . . 13 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
76bnj1196 32674 . . . . . . . . . . . 12 (𝑓𝐶 → ∃𝑑(𝑑𝐵 ∧ dom 𝑓 = 𝑑))
8 bnj1498.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1436 32719 . . . . . . . . . . . . . 14 (𝑑𝐵 → (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
109simpld 494 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
1110anim1i 614 . . . . . . . . . . . 12 ((𝑑𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑𝐴 ∧ dom 𝑓 = 𝑑))
127, 11bnj593 32625 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑(𝑑𝐴 ∧ dom 𝑓 = 𝑑))
13 sseq1 3942 . . . . . . . . . . . 12 (dom 𝑓 = 𝑑 → (dom 𝑓𝐴𝑑𝐴))
1413biimparc 479 . . . . . . . . . . 11 ((𝑑𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓𝐴)
1512, 14bnj593 32625 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑dom 𝑓𝐴)
1615bnj937 32651 . . . . . . . . 9 (𝑓𝐶 → dom 𝑓𝐴)
1716sselda 3917 . . . . . . . 8 ((𝑓𝐶𝑧 ∈ dom 𝑓) → 𝑧𝐴)
1817rexlimiva 3209 . . . . . . 7 (∃𝑓𝐶 𝑧 ∈ dom 𝑓𝑧𝐴)
191, 18sylbi 216 . . . . . 6 (𝑧 𝑓𝐶 dom 𝑓𝑧𝐴)
202bnj1317 32701 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
2120bnj1400 32715 . . . . . 6 dom 𝐶 = 𝑓𝐶 dom 𝑓
2219, 21eleq2s 2857 . . . . 5 (𝑧 ∈ dom 𝐶𝑧𝐴)
23 bnj1498.4 . . . . . 6 𝐹 = 𝐶
2423dmeqi 5802 . . . . 5 dom 𝐹 = dom 𝐶
2522, 24eleq2s 2857 . . . 4 (𝑧 ∈ dom 𝐹𝑧𝐴)
2625ssriv 3921 . . 3 dom 𝐹𝐴
2726a1i 11 . 2 (𝑅 FrSe 𝐴 → dom 𝐹𝐴)
28 bnj1498.2 . . . . . . . 8 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
298, 28, 2bnj1493 32939 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30 vsnid 4595 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
31 elun1 4106 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3230, 31ax-mp 5 . . . . . . . . . 10 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33 eleq2 2827 . . . . . . . . . 10 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
3432, 33mpbiri 257 . . . . . . . . 9 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
3534reximi 3174 . . . . . . . 8 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3635ralimi 3086 . . . . . . 7 (∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
3729, 36syl 17 . . . . . 6 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
38 eliun 4925 . . . . . . 7 (𝑥 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3938ralbii 3090 . . . . . 6 (∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
4037, 39sylibr 233 . . . . 5 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
41 nfcv 2906 . . . . . 6 𝑥𝐴
428bnj1309 32902 . . . . . . . . 9 (𝑡𝐵 → ∀𝑥 𝑡𝐵)
432, 42bnj1307 32903 . . . . . . . 8 (𝑡𝐶 → ∀𝑥 𝑡𝐶)
4443nfcii 2890 . . . . . . 7 𝑥𝐶
45 nfcv 2906 . . . . . . 7 𝑥dom 𝑓
4644, 45nfiun 4951 . . . . . 6 𝑥 𝑓𝐶 dom 𝑓
4741, 46dfss3f 3908 . . . . 5 (𝐴 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
4840, 47sylibr 233 . . . 4 (𝑅 FrSe 𝐴𝐴 𝑓𝐶 dom 𝑓)
4948, 21sseqtrrdi 3968 . . 3 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐶)
5049, 24sseqtrrdi 3968 . 2 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐹)
5127, 50eqssd 3934 1 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2108  {cab 2715  wral 3063  wrex 3064  cun 3881  wss 3883  {csn 4558  cop 4564   cuni 4836   ciun 4921  dom cdm 5580  cres 5582   Fn wfn 6413  cfv 6418   predc-bnj14 32567   FrSe w-bnj15 32571   trClc-bnj18 32573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-bnj17 32566  df-bnj14 32568  df-bnj13 32570  df-bnj15 32572  df-bnj18 32574  df-bnj19 32576
This theorem is referenced by:  bnj60  32942
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