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Theorem bnj1498 31600
Description: Technical lemma for bnj60 31601. 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 4682 . . . . . . 7 (𝑧 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑧 ∈ dom 𝑓)
2 bnj1498.3 . . . . . . . . . . . . . . . 16 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1436 31381 . . . . . . . . . . . . . . 15 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
43bnj1299 31360 . . . . . . . . . . . . . 14 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
5 fndm 6170 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
64, 5bnj31 31259 . . . . . . . . . . . . 13 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
76bnj1196 31336 . . . . . . . . . . . 12 (𝑓𝐶 → ∃𝑑(𝑑𝐵 ∧ dom 𝑓 = 𝑑))
8 bnj1498.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1436 31381 . . . . . . . . . . . . . 14 (𝑑𝐵 → (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
109simpld 488 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
1110anim1i 608 . . . . . . . . . . . 12 ((𝑑𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑𝐴 ∧ dom 𝑓 = 𝑑))
127, 11bnj593 31286 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑(𝑑𝐴 ∧ dom 𝑓 = 𝑑))
13 sseq1 3788 . . . . . . . . . . . 12 (dom 𝑓 = 𝑑 → (dom 𝑓𝐴𝑑𝐴))
1413biimparc 471 . . . . . . . . . . 11 ((𝑑𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓𝐴)
1512, 14bnj593 31286 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑dom 𝑓𝐴)
1615bnj937 31313 . . . . . . . . 9 (𝑓𝐶 → dom 𝑓𝐴)
1716sselda 3763 . . . . . . . 8 ((𝑓𝐶𝑧 ∈ dom 𝑓) → 𝑧𝐴)
1817rexlimiva 3175 . . . . . . 7 (∃𝑓𝐶 𝑧 ∈ dom 𝑓𝑧𝐴)
191, 18sylbi 208 . . . . . 6 (𝑧 𝑓𝐶 dom 𝑓𝑧𝐴)
202bnj1317 31363 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
2120bnj1400 31377 . . . . . 6 dom 𝐶 = 𝑓𝐶 dom 𝑓
2219, 21eleq2s 2862 . . . . 5 (𝑧 ∈ dom 𝐶𝑧𝐴)
23 bnj1498.4 . . . . . 6 𝐹 = 𝐶
2423dmeqi 5495 . . . . 5 dom 𝐹 = dom 𝐶
2522, 24eleq2s 2862 . . . 4 (𝑧 ∈ dom 𝐹𝑧𝐴)
2625ssriv 3767 . . 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 31598 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30 vsnid 4369 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
31 elun1 3944 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3230, 31ax-mp 5 . . . . . . . . . 10 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33 eleq2 2833 . . . . . . . . . 10 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
3432, 33mpbiri 249 . . . . . . . . 9 (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
3534reximi 3157 . . . . . . . 8 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3635ralimi 3099 . . . . . . 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 4682 . . . . . . 7 (𝑥 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3938ralbii 3127 . . . . . 6 (∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
4037, 39sylibr 225 . . . . 5 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
41 nfcv 2907 . . . . . 6 𝑥𝐴
428bnj1309 31561 . . . . . . . . 9 (𝑡𝐵 → ∀𝑥 𝑡𝐵)
432, 42bnj1307 31562 . . . . . . . 8 (𝑡𝐶 → ∀𝑥 𝑡𝐶)
4443nfcii 2898 . . . . . . 7 𝑥𝐶
45 nfcv 2907 . . . . . . 7 𝑥dom 𝑓
4644, 45nfiun 4706 . . . . . 6 𝑥 𝑓𝐶 dom 𝑓
4741, 46dfss3f 3755 . . . . 5 (𝐴 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
4840, 47sylibr 225 . . . 4 (𝑅 FrSe 𝐴𝐴 𝑓𝐶 dom 𝑓)
4948, 21syl6sseqr 3814 . . 3 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐶)
5049, 24syl6sseqr 3814 . 2 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐹)
5127, 50eqssd 3780 1 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  cun 3732  wss 3734  {csn 4336  cop 4342   cuni 4596   ciun 4678  dom cdm 5279  cres 5281   Fn wfn 6065  cfv 6070   predc-bnj14 31228   FrSe w-bnj15 31232   trClc-bnj18 31234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-reg 8708  ax-inf2 8757
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7268  df-1o 7768  df-bnj17 31227  df-bnj14 31229  df-bnj13 31231  df-bnj15 31233  df-bnj18 31235  df-bnj19 31237
This theorem is referenced by:  bnj60  31601
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