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Mirrors > Home > MPE Home > Th. List > acni3 | Structured version Visualization version GIF version |
Description: The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acni3.1 | ⊢ (𝑦 = (𝑔‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
acni3 | ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabn0 3958 | . . . . . 6 ⊢ ({𝑦 ∈ 𝑋 ∣ 𝜑} ≠ ∅ ↔ ∃𝑦 ∈ 𝑋 𝜑) | |
2 | 1 | biimpri 218 | . . . . 5 ⊢ (∃𝑦 ∈ 𝑋 𝜑 → {𝑦 ∈ 𝑋 ∣ 𝜑} ≠ ∅) |
3 | ssrab2 3687 | . . . . 5 ⊢ {𝑦 ∈ 𝑋 ∣ 𝜑} ⊆ 𝑋 | |
4 | 2, 3 | jctil 560 | . . . 4 ⊢ (∃𝑦 ∈ 𝑋 𝜑 → ({𝑦 ∈ 𝑋 ∣ 𝜑} ⊆ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ 𝜑} ≠ ∅)) |
5 | 4 | ralimi 2952 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑 → ∀𝑥 ∈ 𝐴 ({𝑦 ∈ 𝑋 ∣ 𝜑} ⊆ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ 𝜑} ≠ ∅)) |
6 | acni2 8869 | . . 3 ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ({𝑦 ∈ 𝑋 ∣ 𝜑} ⊆ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ 𝜑} ≠ ∅)) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑})) | |
7 | 5, 6 | sylan2 491 | . 2 ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑})) |
8 | acni3.1 | . . . . . . 7 ⊢ (𝑦 = (𝑔‘𝑥) → (𝜑 ↔ 𝜓)) | |
9 | 8 | elrab 3363 | . . . . . 6 ⊢ ((𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑} ↔ ((𝑔‘𝑥) ∈ 𝑋 ∧ 𝜓)) |
10 | 9 | simprbi 480 | . . . . 5 ⊢ ((𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑} → 𝜓) |
11 | 10 | ralimi 2952 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑} → ∀𝑥 ∈ 𝐴 𝜓) |
12 | 11 | anim2i 593 | . . 3 ⊢ ((𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑}) → (𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
13 | 12 | eximi 1762 | . 2 ⊢ (∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ {𝑦 ∈ 𝑋 ∣ 𝜑}) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
14 | 7, 13 | syl 17 | 1 ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 ⟶wf 5884 ‘cfv 5888 AC wacn 8764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-acn 8768 |
This theorem is referenced by: fodomacn 8879 iundom2g 9362 ptclsg 21418 |
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