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Theorem dfac8alem 9443
Description: Lemma for dfac8a 9444. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
Hypotheses
Ref Expression
dfac8alem.2 𝐹 = recs(𝐺)
dfac8alem.3 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))
Assertion
Ref Expression
dfac8alem (𝐴𝐶 → (∃𝑔𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Distinct variable groups:   𝑓,𝑔,𝑦,𝐴   𝐶,𝑔   𝑓,𝐹,𝑦
Allowed substitution hints:   𝐶(𝑦,𝑓)   𝐹(𝑔)   𝐺(𝑦,𝑓,𝑔)

Proof of Theorem dfac8alem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3510 . . 3 (𝐴𝐶𝐴 ∈ V)
2 difss 4105 . . . . . . . . . . . 12 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
3 elpw2g 5238 . . . . . . . . . . . 12 (𝐴 ∈ V → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
42, 3mpbiri 259 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
5 neeq1 3075 . . . . . . . . . . . . 13 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
6 fveq2 6663 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑔𝑦) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
7 id 22 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
86, 7eleq12d 2904 . . . . . . . . . . . . 13 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
95, 8imbi12d 346 . . . . . . . . . . . 12 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
109rspcv 3615 . . . . . . . . . . 11 ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
114, 10syl 17 . . . . . . . . . 10 (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
12113imp 1103 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
13 dfac8alem.2 . . . . . . . . . . . 12 𝐹 = recs(𝐺)
1413tfr2 8023 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
1513tfr1 8022 . . . . . . . . . . . . . 14 𝐹 Fn On
16 fnfun 6446 . . . . . . . . . . . . . 14 (𝐹 Fn On → Fun 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐹
18 vex 3495 . . . . . . . . . . . . 13 𝑥 ∈ V
19 resfunexg 6969 . . . . . . . . . . . . 13 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
2017, 18, 19mp2an 688 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
21 rneq 5799 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐹𝑥) → ran 𝑓 = ran (𝐹𝑥))
22 df-ima 5561 . . . . . . . . . . . . . . . 16 (𝐹𝑥) = ran (𝐹𝑥)
2321, 22syl6eqr 2871 . . . . . . . . . . . . . . 15 (𝑓 = (𝐹𝑥) → ran 𝑓 = (𝐹𝑥))
2423difeq2d 4096 . . . . . . . . . . . . . 14 (𝑓 = (𝐹𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹𝑥)))
2524fveq2d 6667 . . . . . . . . . . . . 13 (𝑓 = (𝐹𝑥) → (𝑔‘(𝐴 ∖ ran 𝑓)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
26 dfac8alem.3 . . . . . . . . . . . . 13 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))
27 fvex 6676 . . . . . . . . . . . . 13 (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2825, 26, 27fvmpt 6761 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
2920, 28ax-mp 5 . . . . . . . . . . 11 (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥)))
3014, 29syl6eq 2869 . . . . . . . . . 10 (𝑥 ∈ On → (𝐹𝑥) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
3130eleq1d 2894 . . . . . . . . 9 (𝑥 ∈ On → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) ↔ (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3212, 31syl5ibrcom 248 . . . . . . . 8 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
33323expia 1113 . . . . . . 7 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))))
3433com23 86 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → (𝑥 ∈ On → ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))))
3534ralrimiv 3178 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
3635ex 413 . . . 4 (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))))
3715tz7.49c 8071 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
3837ex 413 . . . . 5 (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴))
3918f1oen 8518 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴𝑥𝐴)
40 isnumi 9363 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
4139, 40sylan2 592 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥):𝑥1-1-onto𝐴) → 𝐴 ∈ dom card)
4241rexlimiva 3278 . . . . 5 (∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴𝐴 ∈ dom card)
4338, 42syl6 35 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → 𝐴 ∈ dom card))
4436, 43syld 47 . . 3 (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
451, 44syl 17 . 2 (𝐴𝐶 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
4645exlimdv 1925 1 (𝐴𝐶 → (∃𝑔𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cres 5550  cima 5551  Oncon0 6184  Fun wfun 6342   Fn wfn 6343  1-1-ontowf1o 6347  cfv 6348  recscrecs 7996  cen 8494  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-wrecs 7936  df-recs 7997  df-en 8498  df-card 9356
This theorem is referenced by:  dfac8a  9444
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