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Theorem bnj1498 31466
Description: Technical lemma for bnj60 31467. 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 4659 . . . . . . 7 (𝑧 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑧 ∈ dom 𝑓)
2 bnj1498.3 . . . . . . . . . . . . . . . 16 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
32bnj1436 31247 . . . . . . . . . . . . . . 15 (𝑓𝐶 → ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))
43bnj1299 31226 . . . . . . . . . . . . . 14 (𝑓𝐶 → ∃𝑑𝐵 𝑓 Fn 𝑑)
5 fndm 6129 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
64, 5bnj31 31124 . . . . . . . . . . . . 13 (𝑓𝐶 → ∃𝑑𝐵 dom 𝑓 = 𝑑)
76bnj1196 31202 . . . . . . . . . . . 12 (𝑓𝐶 → ∃𝑑(𝑑𝐵 ∧ dom 𝑓 = 𝑑))
8 bnj1498.1 . . . . . . . . . . . . . . 15 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
98bnj1436 31247 . . . . . . . . . . . . . 14 (𝑑𝐵 → (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
109simpld 482 . . . . . . . . . . . . 13 (𝑑𝐵𝑑𝐴)
1110anim1i 602 . . . . . . . . . . . 12 ((𝑑𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑𝐴 ∧ dom 𝑓 = 𝑑))
127, 11bnj593 31152 . . . . . . . . . . 11 (𝑓𝐶 → ∃𝑑(𝑑𝐴 ∧ dom 𝑓 = 𝑑))
13 sseq1 3775 . . . . . . . . . . . 12 (dom 𝑓 = 𝑑 → (dom 𝑓𝐴𝑑𝐴))
1413biimparc 465 . . . . . . . . . . 11 ((𝑑𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓𝐴)
1512, 14bnj593 31152 . . . . . . . . . 10 (𝑓𝐶 → ∃𝑑dom 𝑓𝐴)
1615bnj937 31179 . . . . . . . . 9 (𝑓𝐶 → dom 𝑓𝐴)
1716sselda 3752 . . . . . . . 8 ((𝑓𝐶𝑧 ∈ dom 𝑓) → 𝑧𝐴)
1817rexlimiva 3176 . . . . . . 7 (∃𝑓𝐶 𝑧 ∈ dom 𝑓𝑧𝐴)
191, 18sylbi 207 . . . . . 6 (𝑧 𝑓𝐶 dom 𝑓𝑧𝐴)
202bnj1317 31229 . . . . . . 7 (𝑤𝐶 → ∀𝑓 𝑤𝐶)
2120bnj1400 31243 . . . . . 6 dom 𝐶 = 𝑓𝐶 dom 𝑓
2219, 21eleq2s 2868 . . . . 5 (𝑧 ∈ dom 𝐶𝑧𝐴)
23 bnj1498.4 . . . . . 6 𝐹 = 𝐶
2423dmeqi 5462 . . . . 5 dom 𝐹 = dom 𝐶
2522, 24eleq2s 2868 . . . 4 (𝑧 ∈ dom 𝐹𝑧𝐴)
2625ssriv 3756 . . 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 31464 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30 vsnid 4349 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
31 elun1 3931 . . . . . . . . . . 11 (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
3230, 31ax-mp 5 . . . . . . . . . 10 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33 eleq2 2839 . . . . . . . . . 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 3159 . . . . . . . 8 (∃𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3635ralimi 3101 . . . . . . 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 4659 . . . . . . 7 (𝑥 𝑓𝐶 dom 𝑓 ↔ ∃𝑓𝐶 𝑥 ∈ dom 𝑓)
3938ralbii 3129 . . . . . 6 (∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴𝑓𝐶 𝑥 ∈ dom 𝑓)
4037, 39sylibr 224 . . . . 5 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
41 nfcv 2913 . . . . . 6 𝑥𝐴
428bnj1309 31427 . . . . . . . . 9 (𝑡𝐵 → ∀𝑥 𝑡𝐵)
432, 42bnj1307 31428 . . . . . . . 8 (𝑡𝐶 → ∀𝑥 𝑡𝐶)
4443nfcii 2904 . . . . . . 7 𝑥𝐶
45 nfcv 2913 . . . . . . 7 𝑥dom 𝑓
4644, 45nfiun 4683 . . . . . 6 𝑥 𝑓𝐶 dom 𝑓
4741, 46dfss3f 3744 . . . . 5 (𝐴 𝑓𝐶 dom 𝑓 ↔ ∀𝑥𝐴 𝑥 𝑓𝐶 dom 𝑓)
4840, 47sylibr 224 . . . 4 (𝑅 FrSe 𝐴𝐴 𝑓𝐶 dom 𝑓)
4948, 21syl6sseqr 3801 . . 3 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐶)
5049, 24syl6sseqr 3801 . 2 (𝑅 FrSe 𝐴𝐴 ⊆ dom 𝐹)
5127, 50eqssd 3769 1 (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  cun 3721  wss 3723  {csn 4317  cop 4323   cuni 4575   ciun 4655  dom cdm 5250  cres 5252   Fn wfn 6025  cfv 6030   predc-bnj14 31093   FrSe w-bnj15 31097   trClc-bnj18 31099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-reg 8656  ax-inf2 8705
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7216  df-1o 7716  df-bnj17 31092  df-bnj14 31094  df-bnj13 31096  df-bnj15 31098  df-bnj18 31100  df-bnj19 31102
This theorem is referenced by:  bnj60  31467
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