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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 7120 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑤 ∈ (𝐴 × 𝐵)(𝐹‘𝑤) ∈ 𝐶)) | |
2 | fveq2 6891 | . . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝐹‘⟨𝑥, 𝑦⟩)) | |
3 | df-ov 7414 | . . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩) | |
4 | 2, 3 | eqtr4di 2790 | . . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑤) = (𝑥𝐹𝑦)) |
5 | 4 | eleq1d 2818 | . . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑤) ∈ 𝐶 ↔ (𝑥𝐹𝑦) ∈ 𝐶)) |
6 | 5 | ralxp 5841 | . . 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 274 | 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 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⟨cop 4634 × cxp 5674 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 |
This theorem is referenced by: fovcld 7538 cantnfvalf 9662 axaddf 11142 axmulf 11143 mulnzcnopr 11864 frmdplusg 18771 gass 19206 sylow2blem2 19530 matecl 22147 txdis1cn 23359 isxmet2d 24053 prdsmet 24096 imasdsf1olem 24099 imasf1oxmet 24101 imasf1omet 24102 xmetresbl 24163 comet 24242 tgqioo 24536 xrtgioo 24542 opnmblALT 25344 dvdsmulf1o 26922 hhssabloilem 30769 pstmxmet 33163 xrge0pluscn 33206 isbndx 36953 isbnd3 36955 isbnd3b 36956 prdsbnd 36964 isdrngo2 37129 clintopcllaw 46888 |
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