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Theorem bnj1498 35053
Description: Technical lemma for bnj60 35054. 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 4999 . . . . . . 7 (𝑧 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑧 ∈ dom 𝑓)
2 bnj1498.3 . . . . . . . . . . . . . . . 16 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1436 34831 . . . . . . . . . . . . . . 15 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
43bnj1299 34810 . . . . . . . . . . . . . 14 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
5 fndm 6671 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
64, 5bnj31 34711 . . . . . . . . . . . . 13 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
76bnj1196 34786 . . . . . . . . . . . 12 (𝑓𝐶 → ∃𝑑(𝑑𝐵 ∧ dom 𝑓 = 𝑑))
8 bnj1498.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1436 34831 . . . . . . . . . . . . . 14 (𝑑𝐵 → (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
109simpld 494 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
1110anim1i 615 . . . . . . . . . . . 12 ((𝑑𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑𝐴 ∧ dom 𝑓 = 𝑑))
127, 11bnj593 34737 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑(𝑑𝐴 ∧ dom 𝑓 = 𝑑))
13 sseq1 4020 . . . . . . . . . . . 12 (dom 𝑓 = 𝑑 → (dom 𝑓𝐴𝑑𝐴))
1413biimparc 479 . . . . . . . . . . 11 ((𝑑𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓𝐴)
1512, 14bnj593 34737 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑dom 𝑓𝐴)
1615bnj937 34763 . . . . . . . . 9 (𝑓𝐶 → dom 𝑓𝐴)
1716sselda 3994 . . . . . . . 8 ((𝑓𝐶𝑧 ∈ dom 𝑓) → 𝑧𝐴)
1817rexlimiva 3144 . . . . . . 7 (∃𝑓𝐶 𝑧 ∈ dom 𝑓𝑧𝐴)
191, 18sylbi 217 . . . . . 6 (𝑧 𝑓𝐶 dom 𝑓𝑧𝐴)
202bnj1317 34813 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
2120bnj1400 34827 . . . . . 6 dom 𝐶 = 𝑓𝐶 dom 𝑓
2219, 21eleq2s 2856 . . . . 5 (𝑧 ∈ dom 𝐶𝑧𝐴)
23 bnj1498.4 . . . . . 6 𝐹 = 𝐶
2423dmeqi 5917 . . . . 5 dom 𝐹 = dom 𝐶
2522, 24eleq2s 2856 . . . 4 (𝑧 ∈ dom 𝐹𝑧𝐴)
2625ssriv 3998 . . 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 35051 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30 vsnid 4667 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
31 elun1 4191 . . . . . . . . . . 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 258 . . . . . . . . 9 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
3534reximi 3081 . . . . . . . 8 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3635ralimi 3080 . . . . . . 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 4999 . . . . . . 7 (𝑥 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3938ralbii 3090 . . . . . 6 (∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
4037, 39sylibr 234 . . . . 5 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
41 nfcv 2902 . . . . . 6 𝑥𝐴
428bnj1309 35014 . . . . . . . . 9 (𝑡𝐵 → ∀𝑥 𝑡𝐵)
432, 42bnj1307 35015 . . . . . . . 8 (𝑡𝐶 → ∀𝑥 𝑡𝐶)
4443nfcii 2891 . . . . . . 7 𝑥𝐶
45 nfcv 2902 . . . . . . 7 𝑥dom 𝑓
4644, 45nfiun 5027 . . . . . 6 𝑥 𝑓𝐶 dom 𝑓
4741, 46dfss3f 3986 . . . . 5 (𝐴 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
4840, 47sylibr 234 . . . 4 (𝑅 FrSe 𝐴𝐴 𝑓𝐶 dom 𝑓)
4948, 21sseqtrrdi 4046 . . 3 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐶)
5049, 24sseqtrrdi 4046 . 2 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐹)
5127, 50eqssd 4012 1 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  cun 3960  wss 3962  {csn 4630  cop 4636   cuni 4911   ciun 4995  dom cdm 5688  cres 5690   Fn wfn 6557  cfv 6562   predc-bnj14 34680   FrSe w-bnj15 34684   trClc-bnj18 34686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-bnj17 34679  df-bnj14 34681  df-bnj13 34683  df-bnj15 34685  df-bnj18 34687  df-bnj19 34689
This theorem is referenced by:  bnj60  35054
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