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| 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 10525, allows 𝐵 to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| ac6.1 | ⊢ 𝐴 ∈ V | 
| ac6.2 | ⊢ 𝐵 ∈ V | 
| ac6.3 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| ac6 | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ac6.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | ac6.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | ssrab2 4079 | . . . . . 6 ⊢ {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 | |
| 4 | 3 | rgenw 3064 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 | 
| 5 | iunss 5044 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵) | |
| 6 | 4, 5 | mpbir 231 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ⊆ 𝐵 | 
| 7 | 2, 6 | ssexi 5321 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V | 
| 8 | numth3 10511 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ V → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card) | |
| 9 | 7, 8 | ax-mp 5 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card | 
| 10 | ac6.3 | . . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
| 11 | 10 | ac6num 10520 | . 2 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∈ dom card ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | 
| 12 | 1, 9, 11 | mp3an12 1452 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ⊆ wss 3950 ∪ ciun 4990 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 cardccrd 9976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-ac2 10504 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-en 8987 df-card 9980 df-ac 10157 | 
| This theorem is referenced by: ac6c4 10522 ac6s 10525 wlkiswwlksupgr2 29898 | 
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