![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6859 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶)) | |
2 | fveq2 6645 | . . . . . 6 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝐹‘〈𝑥, 𝑦〉)) | |
3 | df-ov 7138 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘〈𝑥, 𝑦〉) | |
4 | 2, 3 | eqtr4di 2851 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
5 | 4 | eleq1d 2874 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
6 | 5 | ralxp 5676 | . . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) |
7 | 6 | anbi2i 625 | . 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 278 | 1 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 × cxp 5517 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 |
This theorem is referenced by: fovcl 7258 cantnfvalf 9112 axaddf 10556 axmulf 10557 mulnzcnopr 11275 frmdplusg 18011 gass 18423 sylow2blem2 18738 matecl 21030 txdis1cn 22240 isxmet2d 22934 prdsmet 22977 imasdsf1olem 22980 imasf1oxmet 22982 imasf1omet 22983 xmetresbl 23044 comet 23120 tgqioo 23405 xrtgioo 23411 opnmblALT 24207 dvdsmulf1o 25779 hhssabloilem 29044 fovcld 30399 pstmxmet 31250 xrge0pluscn 31293 isbndx 35220 isbnd3 35222 isbnd3b 35223 prdsbnd 35231 isdrngo2 35396 clintopcllaw 44471 |
Copyright terms: Public domain | W3C validator |