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Theorem zorn2lem1 9262
Description: Lemma for zorn2 9272. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Hypotheses
Ref Expression
zorn2lem.3 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
zorn2lem.4 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
zorn2lem.5 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
Assertion
Ref Expression
zorn2lem1 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝐷,𝑓,𝑢,𝑣   𝑓,𝐹,𝑔,𝑢,𝑣,𝑥,𝑧   𝑅,𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶
Allowed substitution hints:   𝐶(𝑥,𝑧,𝑤,𝑢,𝑓,𝑔)   𝐷(𝑥,𝑧,𝑤,𝑔)   𝐹(𝑤)

Proof of Theorem zorn2lem1
StepHypRef Expression
1 zorn2lem.3 . . . . 5 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
21tfr2 7439 . . . 4 (𝑥 ∈ On → (𝐹𝑥) = ((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹𝑥)))
32adantr 481 . . 3 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) = ((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹𝑥)))
41tfr1 7438 . . . . . 6 𝐹 Fn On
5 fnfun 5946 . . . . . 6 (𝐹 Fn On → Fun 𝐹)
64, 5ax-mp 5 . . . . 5 Fun 𝐹
7 vex 3189 . . . . 5 𝑥 ∈ V
8 resfunexg 6433 . . . . 5 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
96, 7, 8mp2an 707 . . . 4 (𝐹𝑥) ∈ V
10 rneq 5311 . . . . . . . . . . . 12 (𝑓 = (𝐹𝑥) → ran 𝑓 = ran (𝐹𝑥))
11 df-ima 5087 . . . . . . . . . . . 12 (𝐹𝑥) = ran (𝐹𝑥)
1210, 11syl6eqr 2673 . . . . . . . . . . 11 (𝑓 = (𝐹𝑥) → ran 𝑓 = (𝐹𝑥))
1312eleq2d 2684 . . . . . . . . . 10 (𝑓 = (𝐹𝑥) → (𝑔 ∈ ran 𝑓𝑔 ∈ (𝐹𝑥)))
1413imbi1d 331 . . . . . . . . 9 (𝑓 = (𝐹𝑥) → ((𝑔 ∈ ran 𝑓𝑔𝑅𝑧) ↔ (𝑔 ∈ (𝐹𝑥) → 𝑔𝑅𝑧)))
1514ralbidv2 2978 . . . . . . . 8 (𝑓 = (𝐹𝑥) → (∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧 ↔ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧))
1615rabbidv 3177 . . . . . . 7 (𝑓 = (𝐹𝑥) → {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧})
17 zorn2lem.4 . . . . . . 7 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
18 zorn2lem.5 . . . . . . 7 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
1916, 17, 183eqtr4g 2680 . . . . . 6 (𝑓 = (𝐹𝑥) → 𝐶 = 𝐷)
2019eleq2d 2684 . . . . . . . 8 (𝑓 = (𝐹𝑥) → (𝑢𝐶𝑢𝐷))
2120imbi1d 331 . . . . . . 7 (𝑓 = (𝐹𝑥) → ((𝑢𝐶 → ¬ 𝑢𝑤𝑣) ↔ (𝑢𝐷 → ¬ 𝑢𝑤𝑣)))
2221ralbidv2 2978 . . . . . 6 (𝑓 = (𝐹𝑥) → (∀𝑢𝐶 ¬ 𝑢𝑤𝑣 ↔ ∀𝑢𝐷 ¬ 𝑢𝑤𝑣))
2319, 22riotaeqbidv 6568 . . . . 5 (𝑓 = (𝐹𝑥) → (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣) = (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣))
24 eqid 2621 . . . . 5 (𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)) = (𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣))
25 riotaex 6569 . . . . 5 (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣) ∈ V
2623, 24, 25fvmpt 6239 . . . 4 ((𝐹𝑥) ∈ V → ((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹𝑥)) = (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣))
279, 26ax-mp 5 . . 3 ((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣))‘(𝐹𝑥)) = (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣)
283, 27syl6eq 2671 . 2 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) = (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣))
29 simprl 793 . . . 4 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝑤 We 𝐴)
30 weso 5065 . . . . . . 7 (𝑤 We 𝐴𝑤 Or 𝐴)
3130ad2antrl 763 . . . . . 6 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
32 vex 3189 . . . . . 6 𝑤 ∈ V
33 soex 7056 . . . . . 6 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
3431, 32, 33sylancl 693 . . . . 5 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝐴 ∈ V)
3518, 34rabexd 4774 . . . 4 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝐷 ∈ V)
36 ssrab2 3666 . . . . . 6 {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧} ⊆ 𝐴
3718, 36eqsstri 3614 . . . . 5 𝐷𝐴
3837a1i 11 . . . 4 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝐷𝐴)
39 simprr 795 . . . 4 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → 𝐷 ≠ ∅)
40 wereu 5070 . . . 4 ((𝑤 We 𝐴 ∧ (𝐷 ∈ V ∧ 𝐷𝐴𝐷 ≠ ∅)) → ∃!𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣)
4129, 35, 38, 39, 40syl13anc 1325 . . 3 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → ∃!𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣)
42 riotacl 6579 . . 3 (∃!𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣 → (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣) ∈ 𝐷)
4341, 42syl 17 . 2 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝑣𝐷𝑢𝐷 ¬ 𝑢𝑤𝑣) ∈ 𝐷)
4428, 43eqeltrd 2698 1 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  ∃!wreu 2909  {crab 2911  Vcvv 3186  wss 3555  c0 3891   class class class wbr 4613  cmpt 4673   Or wor 4994   We wwe 5032  ran crn 5075  cres 5076  cima 5077  Oncon0 5682  Fun wfun 5841   Fn wfn 5842  cfv 5847  crio 6564  recscrecs 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-wrecs 7352  df-recs 7413
This theorem is referenced by:  zorn2lem2  9263  zorn2lem3  9264  zorn2lem4  9265  zorn2lem5  9266
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