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Mirrors > Home > MPE Home > Th. List > Mathboxes > brfvimex | Structured version Visualization version GIF version |
Description: If a binary relation holds and the relation is the value of a function, then the argument to that function is a set. (Contributed by RP, 22-May-2021.) |
Ref | Expression |
---|---|
brfvimex.br | ⊢ (𝜑 → 𝐴𝑅𝐵) |
brfvimex.fv | ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶)) |
Ref | Expression |
---|---|
brfvimex | ⊢ (𝜑 → 𝐶 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brfvimex.fv | . . 3 ⊢ (𝜑 → 𝑅 = (𝐹‘𝐶)) | |
2 | brfvimex.br | . . 3 ⊢ (𝜑 → 𝐴𝑅𝐵) | |
3 | 1, 2 | breqdi 5073 | . 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵) |
4 | brne0 5108 | . 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅) | |
5 | fvprc 6657 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅) | |
6 | 5 | necon1ai 3043 | . 2 ⊢ ((𝐹‘𝐶) ≠ ∅ → 𝐶 ∈ V) |
7 | 3, 4, 6 | 3syl 18 | 1 ⊢ (𝜑 → 𝐶 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 ax-pow 5258 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 |
This theorem is referenced by: ntrclsbex 40377 |
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