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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 6295 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶)) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 6552 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | syl6eqr 2662 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
5 | 4 | eleq1d 2672 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
6 | 5 | ralxp 5185 | . . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) |
7 | 6 | anbi2i 726 | . 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 263 | 1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 |
This theorem is referenced by: fovcl 6663 cantnfvalf 8445 axaddf 9845 axmulf 9846 mulnzcnopr 10552 frmdplusg 17214 gass 17557 sylow2blem2 17859 matecl 20050 txdis1cn 21248 isxmet2d 21942 prdsmet 21985 imasdsf1olem 21988 imasf1oxmet 21990 imasf1omet 21991 xmetresbl 22052 comet 22128 tgqioo 22411 xrtgioo 22417 opnmblALT 23177 dvdsmulf1o 24720 hhssabloilem 27502 fovcld 28820 pstmxmet 29268 xrge0pluscn 29314 isbndx 32751 isbnd3 32753 isbnd3b 32754 prdsbnd 32762 isdrngo2 32927 clintopcllaw 41637 |
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