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Mirrors > Home > MPE Home > Th. List > axcc4 | Structured version Visualization version GIF version |
Description: A version of axcc3 10433 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.) |
Ref | Expression |
---|---|
axcc4.1 | ⊢ 𝐴 ∈ V |
axcc4.2 | ⊢ 𝑁 ≈ ω |
axcc4.3 | ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
axcc4 | ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axcc4.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | rabex 5333 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V |
3 | axcc4.2 | . . 3 ⊢ 𝑁 ≈ ω | |
4 | 2, 3 | axcc3 10433 | . 2 ⊢ ∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) |
5 | rabn0 4386 | . . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) | |
6 | pm2.27 42 | . . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) | |
7 | 5, 6 | sylbir 234 | . . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) |
8 | axcc4.3 | . . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑛) → (𝜑 ↔ 𝜓)) | |
9 | 8 | elrab 3684 | . . . . . . . . 9 ⊢ ((𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓)) |
10 | 7, 9 | imbitrdi 250 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → (({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓))) |
11 | 10 | ral2imi 3086 | . . . . . . 7 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → (∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → ∀𝑛 ∈ 𝑁 ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓))) |
12 | simpl 484 | . . . . . . . 8 ⊢ (((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓) → (𝑓‘𝑛) ∈ 𝐴) | |
13 | 12 | ralimi 3084 | . . . . . . 7 ⊢ (∀𝑛 ∈ 𝑁 ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓) → ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴) |
14 | 11, 13 | syl6 35 | . . . . . 6 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → (∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴)) |
15 | 14 | anim2d 613 | . . . . 5 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) → (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴))) |
16 | ffnfv 7118 | . . . . 5 ⊢ (𝑓:𝑁⟶𝐴 ↔ (𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑓‘𝑛) ∈ 𝐴)) | |
17 | 15, 16 | imbitrrdi 251 | . . . 4 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) → 𝑓:𝑁⟶𝐴)) |
18 | simpr 486 | . . . . . . 7 ⊢ (((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓) → 𝜓) | |
19 | 18 | ralimi 3084 | . . . . . 6 ⊢ (∀𝑛 ∈ 𝑁 ((𝑓‘𝑛) ∈ 𝐴 ∧ 𝜓) → ∀𝑛 ∈ 𝑁 𝜓) |
20 | 11, 19 | syl6 35 | . . . . 5 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → (∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) → ∀𝑛 ∈ 𝑁 𝜓)) |
21 | 20 | adantld 492 | . . . 4 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) → ∀𝑛 ∈ 𝑁 𝜓)) |
22 | 17, 21 | jcad 514 | . . 3 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ((𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) → (𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
23 | 22 | eximdv 1921 | . 2 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑓(𝑓 Fn 𝑁 ∧ ∀𝑛 ∈ 𝑁 ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ → (𝑓‘𝑛) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})) → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓))) |
24 | 4, 23 | mpi 20 | 1 ⊢ (∀𝑛 ∈ 𝑁 ∃𝑥 ∈ 𝐴 𝜑 → ∃𝑓(𝑓:𝑁⟶𝐴 ∧ ∀𝑛 ∈ 𝑁 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∅c0 4323 class class class wbr 5149 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 ωcom 7855 ≈ cen 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7856 df-2nd 7976 df-er 8703 df-en 8940 |
This theorem is referenced by: axcc4dom 10436 supcvg 15802 1stcelcls 22965 iscmet3 24810 ovoliunlem3 25021 itg2seq 25260 nmounbseqi 30030 nmobndseqi 30032 minvecolem5 30134 heibor 36689 |
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