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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnimafnex | Structured version Visualization version GIF version |
Description: The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.) |
Ref | Expression |
---|---|
fnimafnex.f | ⊢ 𝐹 Fn 𝐵 |
Ref | Expression |
---|---|
fnimafnex | ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnimafnex.f | . . 3 ⊢ 𝐹 Fn 𝐵 | |
2 | fnfun 6669 | . . 3 ⊢ (𝐹 Fn 𝐵 → Fun 𝐹) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Fun 𝐹 |
4 | fvex 6920 | . 2 ⊢ (𝐺‘𝐴) ∈ V | |
5 | funimaexg 6654 | . 2 ⊢ ((Fun 𝐹 ∧ (𝐺‘𝐴) ∈ V) → (𝐹 “ (𝐺‘𝐴)) ∈ V) | |
6 | 3, 4, 5 | mp2an 692 | 1 ⊢ (𝐹 “ (𝐺‘𝐴)) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3478 “ cima 5692 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: (None) |
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