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Mirrors > Home > MPE Home > Th. List > fnotovb | Structured version Visualization version GIF version |
Description: Equivalence of operation value and ordered triple membership, analogous to fnopfvb 6935. (Contributed by NM, 17-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 15-Feb-2022.) |
Ref | Expression |
---|---|
fnotovb | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnbrovb 7445 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷〉𝐹𝑅)) | |
2 | df-br 5145 | . . . 4 ⊢ (〈𝐶, 𝐷〉𝐹𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (〈𝐶, 𝐷〉𝐹𝑅 ↔ 〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹)) |
4 | df-ot 4633 | . . . . . 6 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
5 | 4 | eqcomi 2742 | . . . . 5 ⊢ 〈〈𝐶, 𝐷〉, 𝑅〉 = 〈𝐶, 𝐷, 𝑅〉 |
6 | 5 | eleq1i 2825 | . . . 4 ⊢ (〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹) |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (〈〈𝐶, 𝐷〉, 𝑅〉 ∈ 𝐹 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
8 | 1, 3, 7 | 3bitrd 305 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
9 | 8 | 3impb 1116 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 〈cop 4630 〈cotp 4632 class class class wbr 5144 × cxp 5670 Fn wfn 6530 (class class class)co 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-ot 4633 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fn 6538 df-fv 6543 df-ov 7399 |
This theorem is referenced by: (None) |
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