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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 3221 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
2 | abrexss.1 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
3 | 2 | nfcri 2973 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
4 | eleq1 2902 | . . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)) | |
5 | vex 3499 | . . . . 5 ⊢ 𝑧 ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → 𝑧 ∈ V) |
7 | rspa 3208 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
8 | 1, 3, 4, 6, 7 | elabreximd 30272 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) → 𝑧 ∈ 𝐶) |
9 | 8 | ex 415 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → 𝑧 ∈ 𝐶)) |
10 | 9 | ssrdv 3975 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {cab 2801 Ⅎwnfc 2963 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ⊆ wss 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-in 3945 df-ss 3954 |
This theorem is referenced by: funimass4f 30384 measvunilem 31473 |
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