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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > abrexss | Structured version Visualization version GIF version |
Description: A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.) |
Ref | Expression |
---|---|
abrexss.1 | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
abrexss | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 3275 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
2 | abrexss.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | nfcri 2884 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
4 | eleq1 2815 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
5 | vex 3472 | . . . . 5 ⊢ 𝑧 ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → 𝑧 ∈ V) |
7 | rspa 3239 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
8 | 1, 3, 4, 6, 7 | elabreximd 32252 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑧 ∈ 𝐶) |
9 | 8 | ex 412 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑧 ∈ 𝐶)) |
10 | 9 | ssrdv 3983 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cab 2703 Ⅎwnfc 2877 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-v 3470 df-in 3950 df-ss 3960 |
This theorem is referenced by: funimass4f 32366 measvunilem 33740 |
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