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Theorem dfac8alem 9447
Description: Lemma for dfac8a 9448. 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 3517 . . 3 (𝐴𝐶𝐴 ∈ V)
2 difss 4111 . . . . . . . . . . . 12 (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴
3 elpw2g 5243 . . . . . . . . . . . 12 (𝐴 ∈ V → ((𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐹𝑥)) ⊆ 𝐴))
42, 3mpbiri 259 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝐹𝑥)) ∈ 𝒫 𝐴)
5 neeq1 3082 . . . . . . . . . . . . 13 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑦 ≠ ∅ ↔ (𝐴 ∖ (𝐹𝑥)) ≠ ∅))
6 fveq2 6666 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → (𝑔𝑦) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
7 id 22 . . . . . . . . . . . . . 14 (𝑦 = (𝐴 ∖ (𝐹𝑥)) → 𝑦 = (𝐴 ∖ (𝐹𝑥)))
86, 7eleq12d 2911 . . . . . . . . . . . . 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 3621 . . . . . . . . . . 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 1105 . . . . . . . . 9 ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ (𝐴 ∖ (𝐹𝑥)) ≠ ∅) → (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ (𝐴 ∖ (𝐹𝑥)))
13 dfac8alem.2 . . . . . . . . . . . 12 𝐹 = recs(𝐺)
1413tfr2 8028 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐹𝑥) = (𝐺‘(𝐹𝑥)))
1513tfr1 8027 . . . . . . . . . . . . . 14 𝐹 Fn On
16 fnfun 6449 . . . . . . . . . . . . . 14 (𝐹 Fn On → Fun 𝐹)
1715, 16ax-mp 5 . . . . . . . . . . . . 13 Fun 𝐹
18 vex 3502 . . . . . . . . . . . . 13 𝑥 ∈ V
19 resfunexg 6976 . . . . . . . . . . . . 13 ((Fun 𝐹𝑥 ∈ V) → (𝐹𝑥) ∈ V)
2017, 18, 19mp2an 688 . . . . . . . . . . . 12 (𝐹𝑥) ∈ V
21 rneq 5804 . . . . . . . . . . . . . . . 16 (𝑓 = (𝐹𝑥) → ran 𝑓 = ran (𝐹𝑥))
22 df-ima 5566 . . . . . . . . . . . . . . . 16 (𝐹𝑥) = ran (𝐹𝑥)
2321, 22syl6eqr 2878 . . . . . . . . . . . . . . 15 (𝑓 = (𝐹𝑥) → ran 𝑓 = (𝐹𝑥))
2423difeq2d 4102 . . . . . . . . . . . . . 14 (𝑓 = (𝐹𝑥) → (𝐴 ∖ ran 𝑓) = (𝐴 ∖ (𝐹𝑥)))
2524fveq2d 6670 . . . . . . . . . . . . 13 (𝑓 = (𝐹𝑥) → (𝑔‘(𝐴 ∖ ran 𝑓)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
26 dfac8alem.3 . . . . . . . . . . . . 13 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓)))
27 fvex 6679 . . . . . . . . . . . . 13 (𝑔‘(𝐴 ∖ (𝐹𝑥))) ∈ V
2825, 26, 27fvmpt 6764 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ V → (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
2920, 28ax-mp 5 . . . . . . . . . . 11 (𝐺‘(𝐹𝑥)) = (𝑔‘(𝐴 ∖ (𝐹𝑥)))
3014, 29syl6eq 2876 . . . . . . . . . 10 (𝑥 ∈ On → (𝐹𝑥) = (𝑔‘(𝐴 ∖ (𝐹𝑥))))
3130eleq1d 2901 . . . . . . . . 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 1115 . . . . . . 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 3185 . . . . 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 8076 . . . . . 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 8522 . . . . . . 7 ((𝐹𝑥):𝑥1-1-onto𝐴𝑥𝐴)
40 isnumi 9367 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
4139, 40sylan2 592 . . . . . 6 ((𝑥 ∈ On ∧ (𝐹𝑥):𝑥1-1-onto𝐴) → 𝐴 ∈ dom card)
4241rexlimiva 3285 . . . . 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 1927 1 (𝐴𝐶 → (∃𝑔𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3020  wral 3142  wrex 3143  Vcvv 3499  cdif 3936  wss 3939  c0 4294  𝒫 cpw 4541   class class class wbr 5062  cmpt 5142  dom cdm 5553  ran crn 5554  cres 5555  cima 5556  Oncon0 6188  Fun wfun 6345   Fn wfn 6346  1-1-ontowf1o 6350  cfv 6351  recscrecs 8001  cen 8498  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-wrecs 7941  df-recs 8002  df-en 8502  df-card 9360
This theorem is referenced by:  dfac8a  9448
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