![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fnmpo | Structured version Visualization version GIF version |
Description: Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Ref | Expression |
---|---|
fmpo.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
fnmpo | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3464 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
2 | 1 | 2ralimi 3127 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V) |
3 | fmpo.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
4 | 3 | fmpo 8001 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V ↔ 𝐹:(𝐴 × 𝐵)⟶V) |
5 | dffn2 6671 | . . 3 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶V) | |
6 | 4, 5 | bitr4i 278 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V ↔ 𝐹 Fn (𝐴 × 𝐵)) |
7 | 2, 6 | sylib 217 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3065 Vcvv 3446 × cxp 5632 Fn wfn 6492 ⟶wf 6493 ∈ cmpo 7360 |
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 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: fnmpoi 8003 dmmpoga 8006 dmmpogaOLD 8007 fnmpoovd 8020 fsplitfpar 8051 genpdm 10939 isofn 17659 brric 20179 mpocti 31635 f1od2 31641 cnre2csqima 32495 elrnmpoid 43456 smflimlem3 45021 smflimlem6 45024 |
Copyright terms: Public domain | W3C validator |