Theorem ac6 10561

Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 10565, allows 𝐵 to be a proper class. Unlike ac6sg 10569, which allows 𝐵 to be a proper class and so uses the Collection principle bnd2 9959, this needs neither ax-reg 9589 nor ax-inf2 9645. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) Replace hypotheses with antecedents. (Revised by Timothy J. O'Malley, 15-Oct-2026.)

Hypothesis

Ref	Expression
ac6.1	⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ac6	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑥,𝑓,𝐴   𝑦,𝑓,𝐵,𝑥   𝜑,𝑓   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑓)   𝐴(𝑦)   𝑉(𝑥, 𝑦, 𝑓)   𝑊(𝑥, 𝑦, 𝑓)

Proof of Theorem ac6

Step	Hyp	Ref	Expression
1		ssrab2 4005	. . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
2	1	rgenw 3080	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
3		iunss 5002	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵)
4	2, 3	mpbir 234	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵
5		ssexg 5280	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
6	4, 5	mpan 703	. . 3 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V)
7		numth3 10551	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card)
8	6, 7	syl 18	. 2 ⊢ (𝐵 ∈ 𝑊 → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card)
9		ac6.1	. . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
10	9	ac6num 10560	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
11	8, 10	syl3an2 1182	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∪ ciun 4950  dom cdm 5650  ⟶wf 6534  ‘cfv 6538  cardccrd 10019

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-ac2 10544

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-en 8977  df-card 10023  df-ac 10198

This theorem is used by:  ac6c4  10562  ac6s  10565  acsmapd  18733  wlkiswwlksupgr2  30471  foresf1o  33147  ac6mapd  33264  acunirnmpt2  33301  acunirnmpt2f  33302  elrspunidl  34024  fedgmul  34309  reff  34517  cmpcref  34528  omssubadd  34978  ptpconn  36084  nlpfvineqsn  38425  pibt2  38433