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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, 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 6228 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅) | |
3 | 2 | eleq2d 2687 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅)) |
4 | noel 3919 | . . . 4 ⊢ ¬ 𝐴 ∈ ∅ | |
5 | 4 | pm2.21i 116 | . . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V) |
6 | 3, 5 | syl6bi 243 | . 2 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)) |
7 | 1, 6 | pm2.61i 176 | 1 ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 |
This theorem is referenced by: (None) |
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