Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > abfmpel | Structured version Visualization version GIF version |
Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.) |
Ref | Expression |
---|---|
abfmpel.1 | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) |
abfmpel.2 | ⊢ {𝑦 ∣ 𝜑} ∈ V |
abfmpel.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
abfmpel | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abfmpel.2 | . . . . . . 7 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
2 | 1 | csbex 5207 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V |
3 | abfmpel.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑}) | |
4 | 3 | fvmpts 6765 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) |
5 | 2, 4 | mpan2 689 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) |
6 | csbab 4388 | . . . . 5 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑} = {𝑦 ∣ [𝐴 / 𝑥]𝜑} | |
7 | 5, 6 | syl6eq 2872 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = {𝑦 ∣ [𝐴 / 𝑥]𝜑}) |
8 | 7 | eleq2d 2898 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑})) |
9 | 8 | adantr 483 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑})) |
10 | simpl 485 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → 𝐴 ∈ 𝑉) | |
11 | abfmpel.3 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
12 | 11 | ancoms 461 | . . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
13 | 12 | adantll 712 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
14 | 10, 13 | sbcied 3813 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 = 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
15 | 14 | ex 415 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) |
16 | 15 | alrimiv 1924 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) |
17 | elabgt 3662 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦(𝑦 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓)) | |
18 | 16, 17 | sylan2 594 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓)) |
19 | 18 | ancoms 461 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ {𝑦 ∣ [𝐴 / 𝑥]𝜑} ↔ 𝜓)) |
20 | 9, 19 | bitrd 281 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3494 [wsbc 3771 ⦋csb 3882 ↦ cmpt 5138 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 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-iota 6308 df-fun 6351 df-fv 6357 |
This theorem is referenced by: issiga 31366 ismeas 31453 |
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