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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatafv2rnb | 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 the function is defined at 𝐴. (Contributed by AV, 2-Sep-2022.) |
Ref | Expression |
---|---|
dfatafv2rnb | ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funressndmafv2rn 44341 | . 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹) | |
2 | ndfatafv2nrn 44339 | . . . 4 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹) | |
3 | df-nel 3040 | . . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹) | |
4 | 2, 3 | sylib 221 | . . 3 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) ∈ ran 𝐹) |
5 | 4 | con4i 114 | . 2 ⊢ ((𝐹''''𝐴) ∈ ran 𝐹 → 𝐹 defAt 𝐴) |
6 | 1, 5 | impbii 212 | 1 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2110 ∉ wnel 3039 ran crn 5541 defAt wdfat 44234 ''''cafv2 44326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-iota 6327 df-fun 6371 df-dfat 44237 df-afv2 44327 |
This theorem is referenced by: dmafv2rnb 44347 afv2elrn 44349 tz6.12i-afv2 44361 afv2ndeffv0 44378 afv2rnfveq 44380 |
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