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| Mirrors > Home > MPE Home > Th. List > ffnov | Structured version Visualization version GIF version | ||
| Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.) |
| Ref | Expression |
|---|---|
| ffnov | ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ffnfv 7139 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶)) | |
| 2 | fveq2 6906 | . . . . . 6 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝐹‘〈𝑥, 𝑦〉)) | |
| 3 | df-ov 7434 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
| 5 | 4 | eleq1d 2826 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
| 6 | 5 | ralxp 5852 | . . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) |
| 7 | 6 | anbi2i 623 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
| 8 | 1, 7 | bitri 275 | 1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 × cxp 5683 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: fovcld 7560 cantnfvalf 9705 axaddf 11185 axmulf 11186 mulnzcnf 11909 frmdplusg 18867 gass 19319 sylow2blem2 19639 matecl 22431 txdis1cn 23643 isxmet2d 24337 prdsmet 24380 imasdsf1olem 24383 imasf1oxmet 24385 imasf1omet 24386 xmetresbl 24447 comet 24526 tgqioo 24821 xrtgioo 24828 opnmblALT 25638 mpodvdsmulf1o 27237 dvdsmulf1o 27239 hhssabloilem 31280 pstmxmet 33896 xrge0pluscn 33939 mpomulnzcnf 36300 isbndx 37789 isbnd3 37791 isbnd3b 37792 prdsbnd 37800 isdrngo2 37965 aks6d1c6lem3 42173 clintopcllaw 48127 |
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