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Mirrors > Home > ILE Home > Th. List > mpofvexi | GIF version |
Description: Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
fmpo.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
fnmpoi.2 | ⊢ 𝐶 ∈ V |
mpofvexi.3 | ⊢ 𝑅 ∈ V |
mpofvexi.4 | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
mpofvexi | ⊢ (𝑅𝐹𝑆) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpoi.2 | . . 3 ⊢ 𝐶 ∈ V | |
2 | 1 | gen2 1450 | . 2 ⊢ ∀𝑥∀𝑦 𝐶 ∈ V |
3 | mpofvexi.3 | . 2 ⊢ 𝑅 ∈ V | |
4 | mpofvexi.4 | . 2 ⊢ 𝑆 ∈ V | |
5 | fmpo.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
6 | 5 | mpofvex 6203 | . 2 ⊢ ((∀𝑥∀𝑦 𝐶 ∈ V ∧ 𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝐹𝑆) ∈ V) |
7 | 2, 3, 4, 6 | mp3an 1337 | 1 ⊢ (𝑅𝐹𝑆) ∈ V |
Colors of variables: wff set class |
Syntax hints: ∀wal 1351 = wceq 1353 ∈ wcel 2148 Vcvv 2737 (class class class)co 5874 ∈ cmpo 5876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 |
This theorem is referenced by: (None) |
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