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Theorem wfrdmclOLD 8357
Description: Obsolete version of wfrdmcl 8371 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem6OLD.1 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrdmclOLD (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

Proof of Theorem wfrdmclOLD
Dummy variables 𝑓 𝑔 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem6OLD.1 . . . . . . . 8 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
2 dfwrecsOLD 8338 . . . . . . . 8 wrecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
31, 2eqtri 2765 . . . . . . 7 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
43dmeqi 5915 . . . . . 6 dom 𝐹 = dom {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
5 dmuni 5925 . . . . . 6 dom {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔
64, 5eqtri 2765 . . . . 5 dom 𝐹 = 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔
76eleq2i 2833 . . . 4 (𝑋 ∈ dom 𝐹𝑋 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔)
8 eliun 4995 . . . 4 (𝑋 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔 ↔ ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔)
97, 8bitri 275 . . 3 (𝑋 ∈ dom 𝐹 ↔ ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔)
10 eqid 2737 . . . . . . . 8 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
1110wfrlem1OLD 8348 . . . . . . 7 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
1211eqabri 2885 . . . . . 6 (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
13 predeq3 6325 . . . . . . . . . . . . 13 (𝑤 = 𝑋 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑋))
1413sseq1d 4015 . . . . . . . . . . . 12 (𝑤 = 𝑋 → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
1514rspccv 3619 . . . . . . . . . . 11 (∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 → (𝑋𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
1615adantl 481 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → (𝑋𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
17 fndm 6671 . . . . . . . . . . . . 13 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
1817eleq2d 2827 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → (𝑋 ∈ dom 𝑔𝑋𝑧))
1917sseq2d 4016 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))
2018, 19imbi12d 344 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → ((𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) ↔ (𝑋𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)))
2120adantr 480 . . . . . . . . . 10 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → ((𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) ↔ (𝑋𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)))
2216, 21mpbird 257 . . . . . . . . 9 ((𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔))
2322adantrl 716 . . . . . . . 8 ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔))
24233adant3 1133 . . . . . . 7 ((𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔))
2524exlimiv 1930 . . . . . 6 (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧𝐴 ∧ ∀𝑤𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔))
2612, 25sylbi 217 . . . . 5 (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔))
2726reximia 3081 . . . 4 (∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔 → ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)
28 ssiun 5046 . . . 4 (∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔)
2927, 28syl 17 . . 3 (∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔)
309, 29sylbi 217 . 2 (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔)
3130, 6sseqtrrdi 4025 1 (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951   cuni 4907   ciun 4991  dom cdm 5685  cres 5687  Predcpred 6320   Fn wfn 6556  cfv 6561  wrecscwrecs 8336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337
This theorem is referenced by:  wfrlem10OLD  8358  wfrlem14OLD  8362  wfrlem15OLD  8363
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