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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 6721. (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 7207 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷〉𝐹𝑅)) | |
2 | df-br 5069 | . . . 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 4578 | . . . . . 6 ⊢ 〈𝐶, 𝐷, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑅〉 | |
5 | 4 | eqcomi 2832 | . . . . 5 ⊢ 〈〈𝐶, 𝐷〉, 𝑅〉 = 〈𝐶, 𝐷, 𝑅〉 |
6 | 5 | eleq1i 2905 | . . . 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 307 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶𝐹𝐷) = 𝑅 ↔ 〈𝐶, 𝐷, 𝑅〉 ∈ 𝐹)) |
9 | 8 | 3impb 1111 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 〈cotp 4577 class class class wbr 5068 × cxp 5555 Fn wfn 6352 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ov 7161 |
This theorem is referenced by: (None) |
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