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Mirrors > Home > MPE Home > Th. List > ac6s | Structured version Visualization version GIF version |
Description: Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 9053, we derive this strong version of ac6 9637 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.) |
Ref | Expression |
---|---|
ac6s.1 | ⊢ 𝐴 ∈ V |
ac6s.2 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ac6s | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6s.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | bnd2 9053 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑)) |
3 | vex 3401 | . . . . 5 ⊢ 𝑧 ∈ V | |
4 | ac6s.2 | . . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
5 | 1, 3, 4 | ac6 9637 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑 → ∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
6 | 5 | anim2i 610 | . . 3 ⊢ ((𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑) → (𝑧 ⊆ 𝐵 ∧ ∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
7 | 6 | eximi 1878 | . 2 ⊢ (∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
8 | fss 6304 | . . . . . . 7 ⊢ ((𝑓:𝐴⟶𝑧 ∧ 𝑧 ⊆ 𝐵) → 𝑓:𝐴⟶𝐵) | |
9 | 8 | expcom 404 | . . . . . 6 ⊢ (𝑧 ⊆ 𝐵 → (𝑓:𝐴⟶𝑧 → 𝑓:𝐴⟶𝐵)) |
10 | 9 | anim1d 604 | . . . . 5 ⊢ (𝑧 ⊆ 𝐵 → ((𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
11 | 10 | eximdv 1960 | . . . 4 ⊢ (𝑧 ⊆ 𝐵 → (∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
12 | 11 | imp 397 | . . 3 ⊢ ((𝑧 ⊆ 𝐵 ∧ ∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓)) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
13 | 12 | exlimiv 1973 | . 2 ⊢ (∃𝑧(𝑧 ⊆ 𝐵 ∧ ∃𝑓(𝑓:𝐴⟶𝑧 ∧ ∀𝑥 ∈ 𝐴 𝜓)) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
14 | 2, 7, 13 | 3syl 18 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ⊆ wss 3792 ⟶wf 6131 ‘cfv 6135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-reg 8786 ax-inf2 8835 ax-ac2 9620 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-en 8242 df-r1 8924 df-rank 8925 df-card 9098 df-ac 9272 |
This theorem is referenced by: ac6n 9642 ac6s2 9643 ac6sg 9645 ac6sf 9646 nmounbseqiALT 28205 ac6sf2 29994 acunirnmpt2 30025 |
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