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Theorem bnj1498 31255
Description: Technical lemma for bnj60 31256. 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 4556 . . . . . . 7 (𝑧 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑧 ∈ dom 𝑓)
2 bnj1498.3 . . . . . . . . . . . . . . . 16 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1436 31036 . . . . . . . . . . . . . . 15 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
43bnj1299 31015 . . . . . . . . . . . . . 14 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
5 fndm 6028 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
64, 5bnj31 30913 . . . . . . . . . . . . 13 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
76bnj1196 30991 . . . . . . . . . . . 12 (𝑓𝐶 → ∃𝑑(𝑑𝐵 ∧ dom 𝑓 = 𝑑))
8 bnj1498.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1436 31036 . . . . . . . . . . . . . 14 (𝑑𝐵 → (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
109simpld 474 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
1110anim1i 591 . . . . . . . . . . . 12 ((𝑑𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑𝐴 ∧ dom 𝑓 = 𝑑))
127, 11bnj593 30941 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑(𝑑𝐴 ∧ dom 𝑓 = 𝑑))
13 sseq1 3659 . . . . . . . . . . . 12 (dom 𝑓 = 𝑑 → (dom 𝑓𝐴𝑑𝐴))
1413biimparc 503 . . . . . . . . . . 11 ((𝑑𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓𝐴)
1512, 14bnj593 30941 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑dom 𝑓𝐴)
1615bnj937 30968 . . . . . . . . 9 (𝑓𝐶 → dom 𝑓𝐴)
1716sselda 3636 . . . . . . . 8 ((𝑓𝐶𝑧 ∈ dom 𝑓) → 𝑧𝐴)
1817rexlimiva 3057 . . . . . . 7 (∃𝑓𝐶 𝑧 ∈ dom 𝑓𝑧𝐴)
191, 18sylbi 207 . . . . . 6 (𝑧 𝑓𝐶 dom 𝑓𝑧𝐴)
202bnj1317 31018 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
2120bnj1400 31032 . . . . . 6 dom 𝐶 = 𝑓𝐶 dom 𝑓
2219, 21eleq2s 2748 . . . . 5 (𝑧 ∈ dom 𝐶𝑧𝐴)
23 bnj1498.4 . . . . . 6 𝐹 = 𝐶
2423dmeqi 5357 . . . . 5 dom 𝐹 = dom 𝐶
2522, 24eleq2s 2748 . . . 4 (𝑧 ∈ dom 𝐹𝑧𝐴)
2625ssriv 3640 . . 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 31253 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30 vsnid 4242 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
31 elun1 3813 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3230, 31ax-mp 5 . . . . . . . . . 10 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33 eleq2 2719 . . . . . . . . . 10 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
3432, 33mpbiri 248 . . . . . . . . 9 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
3534reximi 3040 . . . . . . . 8 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3635ralimi 2981 . . . . . . 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 4556 . . . . . . 7 (𝑥 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3938ralbii 3009 . . . . . 6 (∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
4037, 39sylibr 224 . . . . 5 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
41 nfcv 2793 . . . . . 6 𝑥𝐴
428bnj1309 31216 . . . . . . . . 9 (𝑡𝐵 → ∀𝑥 𝑡𝐵)
432, 42bnj1307 31217 . . . . . . . 8 (𝑡𝐶 → ∀𝑥 𝑡𝐶)
4443nfcii 2784 . . . . . . 7 𝑥𝐶
45 nfcv 2793 . . . . . . 7 𝑥dom 𝑓
4644, 45nfiun 4580 . . . . . 6 𝑥 𝑓𝐶 dom 𝑓
4741, 46dfss3f 3628 . . . . 5 (𝐴 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
4840, 47sylibr 224 . . . 4 (𝑅 FrSe 𝐴𝐴 𝑓𝐶 dom 𝑓)
4948, 21syl6sseqr 3685 . . 3 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐶)
5049, 24syl6sseqr 3685 . 2 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐹)
5127, 50eqssd 3653 1 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cun 3605  wss 3607  {csn 4210  cop 4216   cuni 4468   ciun 4552  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926   predc-bnj14 30882   FrSe w-bnj15 30886   trClc-bnj18 30888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891
This theorem is referenced by:  bnj60  31256
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