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Theorem zorn2lem4 10537
Description: Lemma for zorn2 10544. (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
zorn2lem4 ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 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 zorn2lem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pm3.24 402 . 2 ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2 df-ne 2939 . . . . 5 (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
32ralbii 3091 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4 df-ral 3060 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5 ralnex 3070 . . . 4 (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
63, 4, 53bitr3i 301 . . 3 (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7 weso 5680 . . . . . . . . 9 (𝑤 We 𝐴𝑤 Or 𝐴)
87adantr 480 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9 vex 3482 . . . . . . . 8 𝑤 ∈ V
10 soex 7944 . . . . . . . 8 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
118, 9, 10sylancl 586 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12 zorn2lem.3 . . . . . . . . . . 11 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
1312tfr1 8436 . . . . . . . . . 10 𝐹 Fn On
14 fvelrnb 6969 . . . . . . . . . 10 (𝐹 Fn On → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹𝑥) = 𝑦))
1513, 14ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹𝑥) = 𝑦)
16 nfv 1912 . . . . . . . . . . 11 𝑥 𝑤 We 𝐴
17 nfa1 2149 . . . . . . . . . . 11 𝑥𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
1816, 17nfan 1897 . . . . . . . . . 10 𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19 nfv 1912 . . . . . . . . . 10 𝑥 𝑦𝐴
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
2120ssrab3 4092 . . . . . . . . . . . . . . . . 17 𝐷𝐴
22 zorn2lem.4 . . . . . . . . . . . . . . . . . 18 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
2312, 22, 20zorn2lem1 10534 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
2421, 23sselid 3993 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐴)
25 eleq1 2827 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2624, 25syl5ibcom 245 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2726exp32 420 . . . . . . . . . . . . . 14 (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑦𝐴))))
2827com12 32 . . . . . . . . . . . . 13 (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → ((𝐹𝑥) = 𝑦𝑦𝐴))))
2928a2d 29 . . . . . . . . . . . 12 (𝑤 We 𝐴 → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3029spsd 2185 . . . . . . . . . . 11 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3130imp 406 . . . . . . . . . 10 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
3218, 19, 31rexlimd 3264 . . . . . . . . 9 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹𝑥) = 𝑦𝑦𝐴))
3315, 32biimtrid 242 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
3433ssrdv 4001 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹𝐴)
3511, 34ssexd 5330 . . . . . 6 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
3635ex 412 . . . . 5 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3736adantl 481 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3812, 22, 20zorn2lem3 10536 . . . . . . . . . . . . . 14 ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
3938exp45 438 . . . . . . . . . . . . 13 (𝑅 Po 𝐴 → (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))))
4039com23 86 . . . . . . . . . . . 12 (𝑅 Po 𝐴 → (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦))))))
4140imp 406 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑤 We 𝐴) → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
4241a2d 29 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))))
4342imp4a 422 . . . . . . . . 9 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4443alrimdv 1927 . . . . . . . 8 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4544alimdv 1914 . . . . . . 7 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
46 r2al 3193 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
4745, 46imbitrrdi 252 . . . . . 6 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)))
48 ssid 4018 . . . . . . . 8 On ⊆ On
4913tz7.48lem 8480 . . . . . . . 8 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
5048, 49mpan 690 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun (𝐹 ↾ On))
51 fnrel 6671 . . . . . . . . . . 11 (𝐹 Fn On → Rel 𝐹)
5213, 51ax-mp 5 . . . . . . . . . 10 Rel 𝐹
5313fndmi 6673 . . . . . . . . . . 11 dom 𝐹 = On
5453eqimssi 4056 . . . . . . . . . 10 dom 𝐹 ⊆ On
55 relssres 6042 . . . . . . . . . 10 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
5652, 54, 55mp2an 692 . . . . . . . . 9 (𝐹 ↾ On) = 𝐹
5756cnveqi 5888 . . . . . . . 8 (𝐹 ↾ On) = 𝐹
5857funeqi 6589 . . . . . . 7 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
5950, 58sylib 218 . . . . . 6 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun 𝐹)
6047, 59syl6 35 . . . . 5 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → Fun 𝐹))
61 onprc 7797 . . . . . 6 ¬ On ∈ V
62 funrnex 7977 . . . . . . . 8 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
6362com12 32 . . . . . . 7 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
64 df-rn 5700 . . . . . . . 8 ran 𝐹 = dom 𝐹
6564eleq1i 2830 . . . . . . 7 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
66 dfdm4 5909 . . . . . . . . 9 dom 𝐹 = ran 𝐹
6753, 66eqtr3i 2765 . . . . . . . 8 On = ran 𝐹
6867eleq1i 2830 . . . . . . 7 (On ∈ V ↔ ran 𝐹 ∈ V)
6963, 65, 683imtr4g 296 . . . . . 6 (Fun 𝐹 → (ran 𝐹 ∈ V → On ∈ V))
7061, 69mtoi 199 . . . . 5 (Fun 𝐹 → ¬ ran 𝐹 ∈ V)
7160, 70syl6 35 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ¬ ran 𝐹 ∈ V))
7237, 71jcad 512 . . 3 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
736, 72biimtrrid 243 . 2 ((𝑅 Po 𝐴𝑤 We 𝐴) → (¬ ∃𝑥 ∈ On 𝐷 = ∅ → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
741, 73mt3i 149 1 ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148  cmpt 5231   Po wpo 5595   Or wor 5596   We wwe 5640  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Rel wrel 5694  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  cfv 6563  crio 7387  recscrecs 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410
This theorem is referenced by:  zorn2lem7  10540
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