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Mirrors > Home > MPE Home > Th. List > Mathboxes > fundmafv2rnb | Structured version Visualization version GIF version |
Description: The alternate function value at a class 𝐴 is defined, i.e., in the range of the function iff 𝐴 is in the domain of the function. (Contributed by AV, 3-Sep-2022.) |
Ref | Expression |
---|---|
fundmafv2rnb | ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funres 6589 | . 2 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴})) | |
2 | dmafv2rnb 46522 | . 2 ⊢ (Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ (𝐹''''𝐴) ∈ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 {csn 4624 dom cdm 5672 ran crn 5673 ↾ cres 5674 Fun wfun 6536 ''''cafv2 46501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6494 df-fun 6544 df-dfat 46412 df-afv2 46502 |
This theorem is referenced by: (None) |
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