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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 7071 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶)) | |
2 | fveq2 6847 | . . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩)) | |
3 | df-ov 7365 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩) | |
4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
5 | 4 | eleq1d 2823 | . . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
6 | 5 | ralxp 5802 | . . 3 ⊢ (∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶) |
7 | 6 | anbi2i 624 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ⟨cop 4597 × cxp 5636 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 |
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-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 df-ov 7365 |
This theorem is referenced by: fovcl 7489 cantnfvalf 9608 axaddf 11088 axmulf 11089 mulnzcnopr 11808 frmdplusg 18671 gass 19088 sylow2blem2 19410 matecl 21790 txdis1cn 23002 isxmet2d 23696 prdsmet 23739 imasdsf1olem 23742 imasf1oxmet 23744 imasf1omet 23745 xmetresbl 23806 comet 23885 tgqioo 24179 xrtgioo 24185 opnmblALT 24983 dvdsmulf1o 26559 hhssabloilem 30245 fovcld 31596 pstmxmet 32518 xrge0pluscn 32561 isbndx 36270 isbnd3 36272 isbnd3b 36273 prdsbnd 36281 isdrngo2 36446 clintopcllaw 46219 |
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