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Theorem zorn2lem4 9656
Description: Lemma for zorn2 9663. (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 pm3.24 393 . 2 ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2 df-ne 2970 . . . . 5 (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
32ralbii 3162 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4 df-ral 3095 . . . 4 (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5 ralnex 3174 . . . 4 (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
63, 4, 53bitr3i 293 . . 3 (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7 weso 5346 . . . . . . . . 9 (𝑤 We 𝐴𝑤 Or 𝐴)
87adantr 474 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9 vex 3401 . . . . . . . 8 𝑤 ∈ V
10 soex 7388 . . . . . . . 8 ((𝑤 Or 𝐴𝑤 ∈ V) → 𝐴 ∈ V)
118, 9, 10sylancl 580 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12 zorn2lem.3 . . . . . . . . . . 11 𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))
1312tfr1 7776 . . . . . . . . . 10 𝐹 Fn On
14 fvelrnb 6503 . . . . . . . . . 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 1957 . . . . . . . . . . 11 𝑥 𝑤 We 𝐴
17 nfa1 2145 . . . . . . . . . . 11 𝑥𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
1816, 17nfan 1946 . . . . . . . . . 10 𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19 nfv 1957 . . . . . . . . . 10 𝑥 𝑦𝐴
20 zorn2lem.5 . . . . . . . . . . . . . . . . . 18 𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}
2120ssrab3 3909 . . . . . . . . . . . . . . . . 17 𝐷𝐴
22 zorn2lem.4 . . . . . . . . . . . . . . . . . 18 𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
2312, 22, 20zorn2lem1 9653 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)
2421, 23sseldi 3819 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐴)
25 eleq1 2847 . . . . . . . . . . . . . . . 16 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐴𝑦𝐴))
2624, 25syl5ibcom 237 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → ((𝐹𝑥) = 𝑦𝑦𝐴))
2726exp32 413 . . . . . . . . . . . . . 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 2171 . . . . . . . . . . 11 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴))))
3130imp 397 . . . . . . . . . 10 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹𝑥) = 𝑦𝑦𝐴)))
3218, 19, 31rexlimd 3208 . . . . . . . . 9 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹𝑥) = 𝑦𝑦𝐴))
3315, 32syl5bi 234 . . . . . . . 8 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹𝑦𝐴))
3433ssrdv 3827 . . . . . . 7 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹𝐴)
3511, 34ssexd 5042 . . . . . 6 ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
3635ex 403 . . . . 5 (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3736adantl 475 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
3812, 22, 20zorn2lem3 9655 . . . . . . . . . . . . . 14 ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))
3938exp45 431 . . . . . . . . . . . . 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 397 . . . . . . . . . . 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 415 . . . . . . . . 9 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4443alrimdv 1972 . . . . . . . 8 ((𝑅 Po 𝐴𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
4544alimdv 1959 . . . . . . 7 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦))))
46 r2al 3121 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) ↔ ∀𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝑥) → ¬ (𝐹𝑥) = (𝐹𝑦)))
4745, 46syl6ibr 244 . . . . . 6 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)))
48 ssid 3842 . . . . . . . 8 On ⊆ On
4913tz7.48lem 7819 . . . . . . . 8 ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦)) → Fun (𝐹 ↾ On))
5048, 49mpan 680 . . . . . . 7 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun (𝐹 ↾ On))
51 fnrel 6234 . . . . . . . . . . 11 (𝐹 Fn On → Rel 𝐹)
5213, 51ax-mp 5 . . . . . . . . . 10 Rel 𝐹
53 fndm 6235 . . . . . . . . . . . 12 (𝐹 Fn On → dom 𝐹 = On)
5413, 53ax-mp 5 . . . . . . . . . . 11 dom 𝐹 = On
5554eqimssi 3878 . . . . . . . . . 10 dom 𝐹 ⊆ On
56 relssres 5687 . . . . . . . . . 10 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
5752, 55, 56mp2an 682 . . . . . . . . 9 (𝐹 ↾ On) = 𝐹
5857cnveqi 5542 . . . . . . . 8 (𝐹 ↾ On) = 𝐹
5958funeqi 6156 . . . . . . 7 (Fun (𝐹 ↾ On) ↔ Fun 𝐹)
6050, 59sylib 210 . . . . . 6 (∀𝑥 ∈ On ∀𝑦𝑥 ¬ (𝐹𝑥) = (𝐹𝑦) → Fun 𝐹)
6147, 60syl6 35 . . . . 5 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → Fun 𝐹))
62 onprc 7262 . . . . . 6 ¬ On ∈ V
63 funrnex 7412 . . . . . . . 8 (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
6463com12 32 . . . . . . 7 (Fun 𝐹 → (dom 𝐹 ∈ V → ran 𝐹 ∈ V))
65 df-rn 5366 . . . . . . . 8 ran 𝐹 = dom 𝐹
6665eleq1i 2850 . . . . . . 7 (ran 𝐹 ∈ V ↔ dom 𝐹 ∈ V)
67 dfdm4 5561 . . . . . . . . 9 dom 𝐹 = ran 𝐹
6854, 67eqtr3i 2804 . . . . . . . 8 On = ran 𝐹
6968eleq1i 2850 . . . . . . 7 (On ∈ V ↔ ran 𝐹 ∈ V)
7064, 66, 693imtr4g 288 . . . . . 6 (Fun 𝐹 → (ran 𝐹 ∈ V → On ∈ V))
7162, 70mtoi 191 . . . . 5 (Fun 𝐹 → ¬ ran 𝐹 ∈ V)
7261, 71syl6 35 . . . 4 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ¬ ran 𝐹 ∈ V))
7337, 72jcad 508 . . 3 ((𝑅 Po 𝐴𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
746, 73syl5bir 235 . 2 ((𝑅 Po 𝐴𝑤 We 𝐴) → (¬ ∃𝑥 ∈ On 𝐷 = ∅ → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
751, 74mt3i 144 1 ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wal 1599   = wceq 1601  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  Vcvv 3398  wss 3792  c0 4141   class class class wbr 4886  cmpt 4965   Po wpo 5272   Or wor 5273   We wwe 5313  ccnv 5354  dom cdm 5355  ran crn 5356  cres 5357  cima 5358  Rel wrel 5360  Oncon0 5976  Fun wfun 6129   Fn wfn 6130  cfv 6135  crio 6882  recscrecs 7750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-wrecs 7689  df-recs 7751
This theorem is referenced by:  zorn2lem7  9659
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