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Mirrors > Home > MPE Home > Th. List > elfv2ex | Structured version Visualization version GIF version |
Description: If a function value of a function value has a member, then the first argument is a set. (Contributed by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
elfv2ex | ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)) | |
2 | fv2prc 6489 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅) | |
3 | 2 | eleq2d 2845 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅)) |
4 | noel 4146 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
5 | 4 | pm2.21i 117 | . . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V) |
6 | 3, 5 | syl6bi 245 | . 2 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)) |
7 | 1, 6 | pm2.61i 177 | 1 ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 ‘cfv 6137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5027 ax-pow 5079 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-dm 5367 df-iota 6101 df-fv 6145 |
This theorem is referenced by: (None) |
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