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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 6951 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅) | |
| 3 | 2 | eleq2d 2827 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅)) | 
| 4 | noel 4338 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
| 5 | 4 | pm2.21i 119 | . . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V) | 
| 6 | 3, 5 | biimtrdi 253 | . 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 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: (None) | 
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