![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6947 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅) | |
3 | 2 | eleq2d 2815 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅)) |
4 | noel 4334 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
5 | 4 | pm2.21i 119 | . . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V) |
6 | 3, 5 | biimtrdi 252 | . 2 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)) |
7 | 1, 6 | pm2.61i 182 | 1 ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2098 Vcvv 3473 ∅c0 4326 ‘cfv 6553 |
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-11 2146 ax-12 2166 ax-ext 2699 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-dm 5692 df-iota 6505 df-fv 6561 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |