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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvopabf4g | Structured version Visualization version GIF version |
Description: Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
fvopabf4g.1 | ⊢ 𝐶 ∈ V |
fvopabf4g.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
fvopabf4g.3 | ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵) |
Ref | Expression |
---|---|
fvopabf4g | ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapg 8784 | . . . 4 ⊢ ((𝑅 ∈ 𝑌 ∧ 𝐷 ∈ 𝑋) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅)) | |
2 | 1 | ancoms 460 | . . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅)) |
3 | 2 | biimp3ar 1471 | . 2 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → 𝐴 ∈ (𝑅 ↑m 𝐷)) |
4 | fvopabf4g.2 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
5 | fvopabf4g.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵) | |
6 | fvopabf4g.1 | . . 3 ⊢ 𝐶 ∈ V | |
7 | 4, 5, 6 | fvmpt 6952 | . 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶) |
8 | 3, 7 | syl 17 | 1 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ↦ cmpt 5192 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ↑m cmap 8771 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 |
This theorem is referenced by: (None) |
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