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Mirrors > Home > MPE Home > Th. List > fvresex | Structured version Visualization version GIF version |
Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvresex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
fvresex | ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3990 | . . . . . . . 8 ⊢ 𝐴 ⊆ V | |
2 | resmpt 5899 | . . . . . . . 8 ⊢ (𝐴 ⊆ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴) = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) |
4 | 3 | fveq1i 6665 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) |
5 | fveq2 6664 | . . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
6 | eqid 2821 | . . . . . . . . 9 ⊢ (𝑧 ∈ V ↦ (𝐹‘𝑧)) = (𝑧 ∈ V ↦ (𝐹‘𝑧)) | |
7 | fvex 6677 | . . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V | |
8 | 5, 6, 7 | fvmpt 6762 | . . . . . . . 8 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥)) |
9 | 8 | elv 3500 | . . . . . . 7 ⊢ ((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥) |
10 | fveqres 6706 | . . . . . . 7 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧))‘𝑥) = (𝐹‘𝑥) → (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥)) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ (((𝑧 ∈ V ↦ (𝐹‘𝑧)) ↾ 𝐴)‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥) |
12 | 4, 11 | eqtr3i 2846 | . . . . 5 ⊢ ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) = ((𝐹 ↾ 𝐴)‘𝑥) |
13 | 12 | eqeq2i 2834 | . . . 4 ⊢ (𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)) |
14 | 13 | exbii 1839 | . . 3 ⊢ (∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥) ↔ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)) |
15 | 14 | abbii 2886 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} = {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} |
16 | fvresex.1 | . . . 4 ⊢ 𝐴 ∈ V | |
17 | 16 | mptex 6978 | . . 3 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) ∈ V |
18 | 17 | fvclex 7651 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧))‘𝑥)} ∈ V |
19 | 15, 18 | eqeltrri 2910 | 1 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∃wex 1771 ∈ wcel 2105 {cab 2799 Vcvv 3495 ⊆ wss 3935 ↦ cmpt 5138 ↾ cres 5551 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: (None) |
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