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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 7757 | . . . 4 ⊢ ((𝑅 ∈ 𝑌 ∧ 𝐷 ∈ 𝑋) → (𝐴 ∈ (𝑅 ↑𝑚 𝐷) ↔ 𝐴:𝐷⟶𝑅)) | |
2 | 1 | ancoms 468 | . . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (𝐴 ∈ (𝑅 ↑𝑚 𝐷) ↔ 𝐴:𝐷⟶𝑅)) |
3 | 2 | biimp3ar 1425 | . 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 6191 | . 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 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: (None) |
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