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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 5156 | . 2 ⊢ (𝜑 → 𝐴(𝐹‘𝐶)𝐵) |
4 | brne0 5191 | . 2 ⊢ (𝐴(𝐹‘𝐶)𝐵 → (𝐹‘𝐶) ≠ ∅) | |
5 | fvprc 6877 | . . 3 ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐶) = ∅) | |
6 | 5 | necon1ai 2962 | . 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 2098 ≠ wne 2934 Vcvv 3468 ∅c0 4317 class class class wbr 5141 ‘cfv 6537 |
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 2163 ax-ext 2697 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 |
This theorem is referenced by: ntrclsbex 43361 |
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