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Theorem dfac8alem 9103
Description: Lemma for dfac8a 9104. 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 3365 . . 3 (𝐴𝐶𝐴 ∈ V)
2 difss 3899 . . . . . . . . . . . 12 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
3 elpw2g 4985 . . . . . . . . . . . 12 (𝐴 ∈ V → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
42, 3mpbiri 249 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
5 neeq1 2999 . . . . . . . . . . . . 13 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
6 fveq2 6375 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑔𝑦) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
7 id 22 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
86, 7eleq12d 2838 . . . . . . . . . . . . 13 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑔𝑦) ∈ 𝑦 ↔ (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
95, 8imbi12d 335 . . . . . . . . . . . 12 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → ((𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ↔ ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))))
109rspcv 3457 . . . . . . . . . . 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 1137 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
13 dfac8alem.2 . . . . . . . . . . . 12 𝐹 = recs(𝐺)
1413tfr2 7698 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
1513tfr1 7697 . . . . . . . . . . . . . 14 𝐹 Fn On
16 fnfun 6166 . . . . . . . . . . . . . 14 (𝐹 Fn On → Fun 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐹
18 vex 3353 . . . . . . . . . . . . 13 𝑥 ∈ V
19 resfunexg 6672 . . . . . . . . . . . . 13 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
2017, 18, 19mp2an 683 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
21 rneq 5519 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐹𝑥) → ran 𝑓 = ran (𝐹𝑥))
22 df-ima 5290 . . . . . . . . . . . . . . . 16 (𝐹𝑥) = ran (𝐹𝑥)
2321, 22syl6eqr 2817 . . . . . . . . . . . . . . 15 (𝑓 = (𝐹𝑥) → ran 𝑓 = (𝐹𝑥))
2423difeq2d 3890 . . . . . . . . . . . . . 14 (𝑓 = (𝐹𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹𝑥)))
2524fveq2d 6379 . . . . . . . . . . . . 13 (𝑓 = (𝐹𝑥) → (𝑔‘(𝐴 ∖ ran 𝑓)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
26 dfac8alem.3 . . . . . . . . . . . . 13 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))
27 fvex 6388 . . . . . . . . . . . . 13 (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2825, 26, 27fvmpt 6471 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
2920, 28ax-mp 5 . . . . . . . . . . 11 (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥)))
3014, 29syl6eq 2815 . . . . . . . . . 10 (𝑥 ∈ On → (𝐹𝑥) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
3130eleq1d 2829 . . . . . . . . 9 (𝑥 ∈ On → ((𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)) ↔ (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥))))
3212, 31syl5ibrcom 238 . . . . . . . 8 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝑥 ∈ On → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
33323expia 1150 . . . . . . 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 3112 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦)) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))))
3635ex 401 . . . 4 (𝐴 ∈ V → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))))
3715tz7.49c 7745 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥)))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
3837ex 401 . . . . 5 (𝐴 ∈ V → (∀𝑥 ∈ On ((𝐴 ∖ (𝐹𝑥)) ≠ ∅ → (𝐹𝑥) ∈ (𝐴 ∖ (𝐹𝑥))) → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴))
3918f1oen 8181 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴𝑥𝐴)
40 isnumi 9023 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
4139, 40sylan2 586 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥):𝑥1-1-onto𝐴) → 𝐴 ∈ dom card)
4241rexlimiva 3175 . . . . 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 2028 1 (𝐴𝐶 → (∃𝑔𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  wss 3732  c0 4079  𝒫 cpw 4315   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Oncon0 5908  Fun wfun 6062   Fn wfn 6063  1-1-ontowf1o 6067  cfv 6068  recscrecs 7671  cen 8157  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-wrecs 7610  df-recs 7672  df-en 8161  df-card 9016
This theorem is referenced by:  dfac8a  9104
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